Introduction and main results
Can averaging a dependent sequence of trace statistics on one random graph produce a law of large numbers and a functional central limit theorem? We study this question for Grover walks on random lifts of a fixed regular base. An N N N N N N
The trace of a power sums contributions from all eigenphases. The expectation of its squared mean-centred trace is the connected spectral form factor. For a non-bipartite base, the squared centred trace divided by its observation time has a nondegenerate limit at a single growing time. Averaging the observation-time-normalised squared traces instead recovers an explicit deterministic value. This question of temporal self-averaging has also been studied for quantum graphs and random-matrix ensembles [4 , 6 , 12 , 15 , 22 , 31 ] .
Our main result describes the cumulative averaging error by a Brownian limit while the number of sheets and the observation window grow simultaneously. The initial time may remain fixed: within the stated logarithmic cutoff range, the window need not be short relative to its initial time. The limit is joint with fixed low-degree polynomial trace statistics and is independent of their limiting Poisson structure. Bipartite and non-bipartite bases have the same averaged limit, but the Brownian variance rate doubles in the bipartite case. Dividing all observations by a common initial time instead leaves a deterministic growth profile and, on proportional windows, a nonconstant variance rate.
Theorem 1.1 gives the functional limit and Theorem 1.2 the law of large numbers. Corollaries 1.3 and 1.4 describe the common-time normalisation. The subsequent fixed-offset theorem identifies the fluctuations that remain at a single time. Both exact mean centring and a centring computable from the base and the number of sheets are available. All trace observables use the full Grover spectrum.
Main contributions. The proof requires uniform estimates because both the number of cycle lengths and the number of observations increase. A fixed-dimensional cycle limit or a stationary Gaussian approximation at fixed time offsets alone would not control the cumulative error. Theorem 4.1 provides a total-variation error of order R 4 q R / N R^4q^R/N R 4 q R / N q = d − 1 q=d-1 q = d − 1 R R R
Model and observables
Let B = ( V , E ) B=(V,E) B = ( V , E ) d d d d ≥ 3 d \ge3 d ≥ 3
q = d − 1 , n 0 = ∣ V ( B ) ∣ , m 0 = ∣ E ( B ) ∣ , r d = 2 q d , α d = arccos r d . q=d-1,\qquad n_0=\lvert V(B)\rvert,\qquad m_0=\lvert E(B)\rvert,\qquad r_d=\frac{2\sqrt{q}}{d},\qquad\alpha_d=\arccos r_d. q = d − 1 , n 0 = ∣ V ( B )∣ , m 0 = ∣ E ( B )∣ , r d = d 2 q , α d = arccos r d . Thus 0 < r d < 1 0<r_d<1 0 < r d < 1 0 < α d < π / 2 0<\alpha_d<\pi/2 0 < α d < π /2 [ N ] = { 1 , … , N } [N]=\{1,\ldots,N\} [ N ] = { 1 , … , N } e = ( u , v ) e=(u,v) e = ( u , v ) σ e \sigma_e σ e ( u , a ) (u,a) ( u , a ) ( v , σ e ( a ) ) (v,\sigma_e(a)) ( v , σ e ( a )) a ∈ [ N ] a\in[N] a ∈ [ N ] N N N G N G_N G N S N \mathfrak{S}_N S N [ N ] [N] [ N ] C B C_B C B
For a finite simple d d d G G G A G A_G A G D ( G ) D(G) D ( G ) o ( e ) , t ( e ) o(e),t(e) o ( e ) , t ( e ) e ˉ \bar e e ˉ I n I_n I n n × n n\times n n × n 1 \mathbf{1} 1 M T M^{\mathsf T} M T M ∗ M^* M ∗
( U G ) e , f = 2 d 1 { t ( f ) = o ( e ) } − 1 { e = f ˉ } . (U_G)_{e,f}=\frac{2}{d}\mathbf{1}_{\{t(f)=o(e)\}}-\mathbf{1}_{\{e=\bar f\}}. ( U G ) e , f = d 2 1 { t ( f ) = o ( e )} − 1 { e = f ˉ } . It is real and orthogonal: it is the product of the arc-reversal involution and the direct sum of the local reflections ( 2 / d ) 11 ∗ − I d (2/d)\mathbf{1}\mathbf{1}^{*}-I_d ( 2/ d ) 1 1 ∗ − I d d d d
A N ( s ) = Tr U G N s − E [ Tr U G N s ] , s ∈ Z ≥ 0 , \mathcal{A}_N(s)=\operatorname{Tr}U_{G_N}^{s}-\mathbb{E}[\operatorname{Tr}U_{G_N}^{s}],\qquad s\in\mathbb{Z}_{\geq0}, A N ( s ) = Tr U G N s − E [ Tr U G N s ] , s ∈ Z ≥ 0 , and the connected form factor with no matrix-dimension normalisation,
F N c ( s ) = E [ ∣ A N ( s ) ∣ 2 ] = E [ ∣ Tr U G N s ∣ 2 ] − ∣ E [ Tr U G N s ] ∣ 2 . \mathcal{F}^{\mathrm{c}}_N(s)=\mathbb{E}[|\mathcal{A}_N(s)|^2]=\mathbb{E}[|\operatorname{Tr}U_{G_N}^{s}|^2]-|\mathbb{E}[\operatorname{Tr}U_{G_N}^{s}]|^2. F N c ( s ) = E [ ∣ A N ( s ) ∣ 2 ] = E [ ∣ Tr U G N s ∣ 2 ] − ∣ E [ Tr U G N s ] ∣ 2 . The symbols P \mathbb{P} P E \mathbb{E} E G N G_N G N
For any graph G G G H G H_G H G
( H B ) e , f = 1 { t ( e ) = o ( f ) , f ≠ e ˉ } . (H_B)_{e,f}=\mathbf{1}\{t(e)=o(f),\ f\ne\bar{e}\}. ( H B ) e , f = 1 { t ( e ) = o ( f ) , f = e ˉ } . The trace Tr H G k \operatorname{Tr}H_G^k Tr H G k H B H_B H B B B B Lemma 7.1 . The time-averaging results below cover both bipartite and non-bipartite bases. Define
Γ d ( h ) = 2 π ∫ α d π − α d cos ( h θ ) d θ , h ∈ Z . (1) \Gamma_d(h)=\frac{2}{\pi}\int_{\alpha_d}^{\pi-\alpha_d}\cos(h\theta)\,\mathrm{d}\theta,\qquad h\in\mathbb{Z}. \tag*{(1)} Γ d ( h ) = π 2 ∫ α d π − α d cos ( h θ ) d θ , h ∈ Z . (1) The variance coefficient and the period are
c d = Γ d ( 0 ) = 4 π arcsin ( 2 d − 1 d ) , p B = { 1 , B is not bipartite , 2 , B is bipartite . c_d=\Gamma_d(0)=\frac{4}{\pi}\arcsin\left(\frac{2\sqrt{d-1}}{d}\right),\qquad
p_B=\begin{cases}
1,& B\text{ is not bipartite},\\
2,& B\text{ is bipartite}.
\end{cases} c d = Γ d ( 0 ) = π 4 arcsin ( d 2 d − 1 ) , p B = { 1 , 2 , B is not bipartite , B is bipartite . The set I d = [ α d , π − α d ] ∪ [ − π + α d , − α d ] I_d=[\alpha_d,\pi-\alpha_d]\cup[-\pi+\alpha_d,-\alpha_d] I d = [ α d , π − α d ] ∪ [ − π + α d , − α d ] Γ d \Gamma_d Γ d π − 1 1 I d \pi^{-1}\mathbf{1}_{I_d} π − 1 1 I d c d = ∣ I d ∣ / π = 2 ∣ I d ∣ / ( 2 π ) c_d=|I_d|/\pi=2|I_d|/(2\pi) c d = ∣ I d ∣/ π = 2∣ I d ∣/ ( 2 π ) 2 2 2 Lemma 7.1 , p B p_B p B H B H_B H B R R R
τ N , R = R 4 q R / N , δ N , R = R 5 q R / N . \tau_{N,R}=R^4q^R/N,\qquad\delta_{N,R}=R^5q^R/N. τ N , R = R 4 q R / N , δ N , R = R 5 q R / N . The quantity τ N , R \tau_{N,R} τ N , R δ N , R \delta_{N,R} δ N , R Y N , k Y_{N,k} Y N , k Y k Y_k Y k § 2 . We use W W W
Time averaging on one lift
We distinguish two choices of centring and two choices of time normalisation. Set
A N 0 ( s ) = A N ( s ) , A N ∗ ( s ) = Tr U G N s − m N ( s ) , \mathcal{A}_N^0(s)=\mathcal{A}_N(s),\qquad
\mathcal{A}_N^*(s)=\operatorname{Tr}U_{G_N}^{s}-m_N(s), A N 0 ( s ) = A N ( s ) , A N ∗ ( s ) = Tr U G N s − m N ( s ) , where m N ( s ) m_N(s) m N ( s ) § 9.1 . The superscripts 0 0 0 ∗ * ∗ 0 0 0 ∗ * ∗ ⋆ ∈ { 0 , ∗ } \star\in\{0,*\} ⋆ ∈ { 0 , ∗ }
F ^ N , t , M ♮ , ⋆ = 1 M ∑ h = 0 M − 1 A N ⋆ ( t + h ) 2 t + h , F ^ N , t , M ⋆ = 1 M t ∑ h = 0 M − 1 A N ⋆ ( t + h ) 2 . \widehat{\mathcal{F}}_{N,t,M}^{\natural,\star}
=\frac{1}{M}\sum_{h=0}^{M-1}\frac{\mathcal{A}_N^\star(t+h)^2}{t+h},
\qquad
\widehat{\mathcal{F}}_{N,t,M}^{\star}
=\frac{1}{Mt}\sum_{h=0}^{M-1}\mathcal{A}_N^\star(t+h)^2. F N , t , M ♮ , ⋆ = M 1 h = 0 ∑ M − 1 t + h A N ⋆ ( t + h ) 2 , F N , t , M ⋆ = M t 1 h = 0 ∑ M − 1 A N ⋆ ( t + h ) 2 . Every summand is evaluated on the same random lift. The symbol ♮ means that each squared trace is divided by its own observation time. The cumulative error for this normalisation is
Z N , t , M ♮ , ⋆ ( u ) = 1 M ∑ h = 0 ⌊ M u ⌋ − 1 { A ^ N ⋆ ( t + h ) 2 t + h − c d } , 0 ≤ u ≤ 1 , \mathcal{Z}^{\natural,\star}_{N,t,M}(u)=\frac{1}{\sqrt{M}}\sum_{h=0}^{\lfloor Mu\rfloor-1}\left\{\frac{\widehat{A}^{\star}_{N}(t+h)^{2}}{t+h}-c_d\right\},\qquad0\le u\le1, Z N , t , M ♮ , ⋆ ( u ) = M 1 h = 0 ∑ ⌊ M u ⌋ − 1 { t + h A N ⋆ ( t + h ) 2 − c d } , 0 ≤ u ≤ 1 , Empty sums are zero. Each process is right-continuous and constant between consecutive points u = j / M u=j/M u = j / M
Theorem 1.1 (Fluctuations with observation-time normalisation). Let B B B d d d d ≥ 3 d\ge3 d ≥ 3 t N t_N t N M N M_N M N
t N ≥ 1 , M N ⟶ ∞ , R N = t N + M N , τ N , R N ⟶ 0. (2) t_N\ge1,\qquad M_N\longrightarrow\infty,\qquad R_N=t_N+M_N,\qquad\tau_{N,R_N}\longrightarrow0. \tag*{(2)} t N ≥ 1 , M N ⟶ ∞ , R N = t N + M N , τ N , R N ⟶ 0. (2) Then, for either trace centring,
Z N , t N , M N ♮ , ⋆ ⟹ 2 p B c d W in D ( [ 0 , 1 ] ) with J 1 . \mathcal{Z}^{\natural,\star}_{N,t_N,M_N}\Longrightarrow2\sqrt{p_Bc_d}\,W\quad\text{in }D([0,1])\text{ with }J_1. Z N , t N , M N ♮ , ⋆ ⟹ 2 p B c d W in D ([ 0 , 1 ]) with J 1 . The polygonal interpolations converge in C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) L ≥ 3 L\ge3 L ≥ 3 ( Y N , k ) k = 3 L (Y_{N,k})_{k=3}^{L} ( Y N , k ) k = 3 L ( Y k ) k = 3 L (Y_k)_{k=3}^{L} ( Y k ) k = 3 L W W W
M N ( F ^ N , t N , M N ♮ , ⋆ − c d ) ⟹ N ( 0 , 4 p B c d ) . \sqrt{M_N}\left(\widehat{F}^{\natural,\star}_{N,t_N,M_N}-c_d\right)\Longrightarrow\mathcal{N}(0,4p_Bc_d). M N ( F N , t N , M N ♮ , ⋆ − c d ) ⟹ N ( 0 , 4 p B c d ) . Apart from the common cutoff condition, Theorem 1.1 imposes no restriction on the ratio M N / t N M_N/t_N M N / t N t N t_N t N t N + M N ≤ ( 1 − ε ) log q N t_N+M_N\le(1-\varepsilon)\log_q N t N + M N ≤ ( 1 − ε ) log q N 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 t N = 1 t_N=1 t N = 1 M N = ⌊ ( 1 − ε ) log q N ⌋ − 1 M_N=\lfloor(1-\varepsilon)\log_q N\rfloor-1 M N = ⌊( 1 − ε ) log q N ⌋ − 1 N N N
The law of large numbers distinguishes the two normalisations. Dividing each summand by its own observation time recovers the same constant on all the preceding windows. With a common initial time, the limiting average also records deterministic growth across the window.
Theorem 1.2 (Self-averaging for all regular bases). Under (1.2), for either trace centring,
F ^ N , t N , M N ♮ , ⋆ ⟶ c d in probability . \widehat{F}^{\natural,\star}_{N,t_N,M_N}\longrightarrow c_d\quad\text{in probability}. F N , t N , M N ♮ , ⋆ ⟶ c d in probability . If, in addition, M N / t N → ρ ∈ [ 0 , ∞ ) M_N/t_N\to\rho\in[0,\infty) M N / t N → ρ ∈ [ 0 , ∞ )
F ^ N , t N , M N ⋆ ⟶ c d ( 1 + ρ / 2 ) in probability . \widehat{F}^{\star}_{N,t_N,M_N}\longrightarrow c_d(1+\rho/2)\quad\text{in probability}. F N , t N , M N ⋆ ⟶ c d ( 1 + ρ /2 ) in probability . The respective limits hold in L 1 L^1 L 1
δ N , R N M N = R N 5 q R N M N ⟶ 0. (3) \frac{\delta_{N,R_N}}{M_N}=\frac{R_N^5q^{R_N}}{M_N}\longrightarrow0. \tag*{(3)} M N δ N , R N = M N R N 5 q R N ⟶ 0. (3) If R N / M N R_N/M_N R N / M N τ N , R N → 0 \tau_{N,R_N}\to0 τ N , R N → 0 M N / t N M_N/t_N M N / t N
The drift caused by a common time normalisation
Dividing all squared traces by the initial time leaves a deterministic growth term across the window. To compare with the preceding process, define
Z N , t , M ⋆ ( u ) = 1 M ∑ h = 0 ⌊ M u ⌋ − 1 { A N ⋆ ( t + h ) 2 t − c d } , 0 ≤ u ≤ 1. \mathcal{Z}^{\star}_{N,t,M}(u)=\frac{1}{\sqrt{M}}\sum_{h=0}^{\lfloor Mu\rfloor-1}\left\{\frac{A_N^{\star}(t+h)^2}{t}-c_d\right\},\qquad0\leq u\leq1. Z N , t , M ⋆ ( u ) = M 1 h = 0 ∑ ⌊ M u ⌋ − 1 { t A N ⋆ ( t + h ) 2 − c d } , 0 ≤ u ≤ 1. We denote its polygonal interpolation by Z ~ N , t , M ⋆ \widetilde{\mathcal{Z}}^{\star}_{N,t,M} Z N , t , M ⋆
For windows comparable to the initial time, subtracting the time-dependent profile gives the following functional limit.
Corollary 1.3 (Common-time profile and Brownian time change). Assume (1.2) and M N / t N → ρ ∈ [ 0 , ∞ ) M_N/t_N\to\rho\in[0,\infty) M N / t N → ρ ∈ [ 0 , ∞ ) and put σ B = 2 p B c d \sigma_B=2\sqrt{p_Bc_d} σ B = 2 p B c d For either trace centring,
1 M N ∑ h < ⌊ M N u ⌋ { A N ⋆ ( t N + h ) 2 t N − c d t N + h t N } ⟹ σ B ∫ 0 u ( 1 + ρ v ) d W ( v ) . (4) \frac{1}{\sqrt{M_N}}\sum_{h<\lfloor M_Nu\rfloor}\left\{\frac{A_N^{\star}(t_N+h)^2}{t_N}-c_d\frac{t_N+h}{t_N}\right\}\Longrightarrow\sigma_B\int_0^u(1+\rho v)\,\mathrm{d}W(v). \tag*{(4)} M N 1 h < ⌊ M N u ⌋ ∑ { t N A N ⋆ ( t N + h ) 2 − c d t N t N + h } ⟹ σ B ∫ 0 u ( 1 + ρ v ) d W ( v ) . (4) Convergence holds in D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) with J 1 J_1 J 1 and the polygonal interpolations converge in C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) Jointly, the observation-time process in Theorem 1.1 converges to σ B W \sigma_BW σ B W with the same Brownian motion. The two trace centrings have the same joint limit, and the limiting processes are independent of each fixed low-mode vector. The limit in (4) has the same law as
u ⟼ σ B W ~ ( u + ρ u 2 + ρ 2 u 3 3 ) , u\longmapsto\sigma_B\widetilde{W}\left(u+\rho u^2+\frac{\rho^2u^3}{3}\right), u ⟼ σ B W ( u + ρ u 2 + 3 ρ 2 u 3 ) , where $\widetilde{W}) is standard Brownian motion. In particular,
M N { F ^ N , t N , M N ⋆ − c d ( 1 + M N − 1 2 t N ) } ⟹ N ( 0 , 4 p B c d ( 1 + ρ + ρ 2 3 ) ) . \sqrt{M_N}\left\{\widehat{F}^{\star}_{N,t_N,M_N}-c_d\left(1+\frac{M_N-1}{2t_N}\right)\right\}\Longrightarrow\mathcal{N}\left(0,4p_Bc_d\left(1+\rho+\frac{\rho^2}{3}\right)\right). M N { F N , t N , M N ⋆ − c d ( 1 + 2 t N M N − 1 ) } ⟹ N ( 0 , 4 p B c d ( 1 + ρ + 3 ρ 2 ) ) . On windows with M N / t N → 0 M_N/t_N\to0 M N / t N → 0 c d c_d c d M N 3 / 2 / t N M_N^{3/2}/t_N M N 3/2 / t N
Corollary 1.4 (Drift under the common initial-time normalisation). Under (1.2) and M N / t N → 0 M_N/t_N\to0 M N / t N → 0 for either centring,
∥ Z ~ N , t N , M N ⋆ − Z ~ N , t N , M N ♭ , ⋆ − c d M N 3 / 2 2 t N ( u ↦ u 2 ) ∥ ∞ ⟶ 0 (5) \left\|\widetilde{\mathcal{Z}}^{\star}_{N,t_N,M_N}-\widetilde{\mathcal{Z}}^{\flat,\star}_{N,t_N,M_N}-\frac{c_dM_N^{3/2}}{2t_N}(u\mapsto u^2)\right\|_{\infty}\longrightarrow0 \tag*{(5)} Z N , t N , M N ⋆ − Z N , t N , M N ♭ , ⋆ − 2 t N c d M N 3/2 ( u ↦ u 2 ) ∞ ⟶ 0 (5) in probability.
If M N 3 / 2 / t N → κ ∈ [ 0 , ∞ ) M_N^{3/2}/t_N\to\kappa\in[0,\infty) M N 3/2 / t N → κ ∈ [ 0 , ∞ ) the common-time process and its interpolation converge to
u ⟼ 2 p B c d W ~ ( u ) + κ c d 2 u 2 . u\longmapsto2\sqrt{p_Bc_d}\,\widetilde{W}(u)+\frac{\kappa c_d}{2}u^2. u ⟼ 2 p B c d W ( u ) + 2 κ c d u 2 . If M N 3 / 2 / t N → ∞ M_N^{3/2}/t_N\to\infty M N 3/2 / t N → ∞ its terminal value tends to + ∞ +\infty + ∞ in probability, so the sequence is not tight. The finite- κ \kappa κ limit holds jointly with the same independent low-mode vector as in Theorem 1.1 .
For power windows M N = ⌊ t N γ ⌋ M_N=\lfloor t_N^\gamma\rfloor M N = ⌊ t N γ ⌋ 0 < γ < 1 0<\gamma<1 0 < γ < 1 γ < 2 / 3 \gamma<2/3 γ < 2/3 c d u 2 / 2 c_du^2/2 c d u 2 /2 γ = 2 / 3 \gamma=2/3 γ = 2/3 γ > 2 / 3 \gamma>2/3 γ > 2/3
Table 1 compares the three cumulative statistics. For either trace centring, the process in each row is the sum of the indicated summands over 0 ≤ h < ⌊ M N u ⌋ 0\leq h<\lfloor M_Nu\rfloor 0 ≤ h < ⌊ M N u ⌋ M N \sqrt{M_N} M N Q N , h = A N ⋆ ( t N + h ) 2 Q_{N,h}=A_N^{\star}(t_N+h)^2 Q N , h = A N ⋆ ( t N + h ) 2 σ B = 2 p B c d \sigma_B=2\sqrt{p_Bc_d} σ B = 2 p B c d
Centring each squared observation at its own expectation is a separate operation from either choice of trace centring. Corollary 10.6 gives expectation-centred versions of the first two limits in Table 1 under the additional sufficient condition δ N , R N / M N → 0 \delta_{N,R_N}/\sqrt{M_N}\to0 δ N , R N / M N → 0 M N / t N → ρ < ∞ M_N/t_N\to\rho<\infty M N / t N → ρ < ∞ M N 3 / 2 / t N M_N^{3/2}/t_N M N 3/2 / t N
Fluctuations at nearby times
We now turn from time averages to observations at fixed time differences. The next theorem describes their joint fluctuations for a non-bipartite base. The covariance depends on the base only through its degree. The corresponding parity-dependent statement for bipartite bases is Theorem 10.4 .
Theorem 1.5 (Grover trace fluctuations at growing times). Suppose H B H_B H B Fix distinct integers h 1 , … , h p h_1,\ldots,h_p h 1 , … , h p H = max a ∣ h a ∣ H=\max_a|h_a| H = max a ∣ h a ∣ t N t_N t N R N = t N + H R_N=t_N+H R N = t N + H
R N 4 q R N N ⟶ 0 , (6) \frac{R_N^4q^{R_N}}{N}\longrightarrow0, \tag*{(6)} N R N 4 q R N ⟶ 0 , (6) then
( A N ( t N + h a ) t N ) a = 1 p → d ( G h a ) a = 1 p , E [ G h G h ′ ] = Γ d ( h − h ′ ) , (7) \left(\frac{A_N(t_N+h_a)}{\sqrt{t_N}}\right)_{a=1}^p \xrightarrow{\mathrm{d}} (G_{h_a})_{a=1}^p,\qquad\mathbb{E}[G_hG_{h'}]=\Gamma_d(h-h'), \tag*{(7)} ( t N A N ( t N + h a ) ) a = 1 p d ( G h a ) a = 1 p , E [ G h G h ′ ] = Γ d ( h − h ′ ) , (7) where ( G h ) h ∈ Z (G_h)_{h\in\mathbb{Z}} ( G h ) h ∈ Z
R N 5 q R N N ⟶ 0 , (8) \frac{R_N^5q^{R_N}}{N}\longrightarrow0, \tag*{(8)} N R N 5 q R N ⟶ 0 , (8) then the convergence in (7) is in quadratic Wasserstein distance, and every covariance converges. For each fixed L ≥ 3 L\ge3 L ≥ 3 ( Y N , k ) k = 3 L (Y_{N,k})_{k=3}^L ( Y N , k ) k = 3 L ( ( G h a ) a = 1 p , ( Y k ) k = 3 L ) ((G_{h_a})_{a=1}^p,(Y_k)_{k=3}^L) (( G h a ) a = 1 p , ( Y k ) k = 3 L ) Y N , k Y_{N,k} Y N , k Y k Y_k Y k § 2 . This joint convergence is weak under (6) and holds in quadratic Wasserstein distance under (8) .
For every fixed 0 < ε < 1 0<\varepsilon<1 0 < ε < 1
R N ≤ ( 1 − ε ) log q N R_N\le(1-\varepsilon)\log_q N R N ≤ ( 1 − ε ) log q N is sufficient for both (6) and (8) . These sufficient conditions differ near the logarithmic cutoff; neither is claimed to be optimal. Theorem 1.5 involves no time averaging.
Corollary 1.6 (Linear connected form factor). Under the assumptions of Theorem 1.5 and
t N 5 q t N N ⟶ 0 , \frac{t_N^5q^{t_N}}{N}\longrightarrow0, N t N 5 q t N ⟶ 0 , F N c ( t N ) t N ⟶ c d = Γ d ( 0 ) = 4 π arcsin ( 2 d − 1 d ) . (9) \frac{\mathcal{F}^{\mathrm{c}}_N(t_N)}{t_N}\longrightarrow c_d=\Gamma_d(0)=\frac{4}{\pi}\arcsin\left(\frac{2\sqrt{d-1}}{d}\right). \tag*{(9)} t N F N c ( t N ) ⟶ c d = Γ d ( 0 ) = π 4 arcsin ( d 2 d − 1 ) . (9) More generally, the normalised connected trace covariance at fixed offsets converges to the kernel in (1) .
The parity formulas are
Γ d ( 2 m ) = − 2 sin ( 2 m α d ) π m ( m ∈ Z ∖ { 0 } ) , Γ d ( 2 m + 1 ) = 0 ( m ∈ Z ) . (10) \Gamma_d(2m)=-\frac{2\sin(2m\alpha_d)}{\pi m}\quad(m\in\mathbb{Z}\setminus\{0\}),\\
\Gamma_d(2m+1)=0\quad(m\in\mathbb{Z}). \tag*{(10)} Γ d ( 2 m ) = − π m 2 sin ( 2 m α d ) ( m ∈ Z ∖ { 0 }) , Γ d ( 2 m + 1 ) = 0 ( m ∈ Z ) . (10) The even and odd subsequences of the limiting Gaussian process are therefore independent, while successive coordinates within either subsequence need not be independent. In particular, Γ d ( 2 ) < 0 \Gamma_d(2)<0 Γ d ( 2 ) < 0
Outline of the proof
Trace identities express the observable in terms of cyclically non-backtracking closed walks. Simple cycles supply the main random contribution; the remaining walks are controlled by direct moment bounds. Since the number of relevant cycle lengths grows with time, Theorem 4.1 approximates the whole vector up to the terminal cutoff rather than only finitely many fixed lengths.
The analytic step is uniform in both observation times. Lemma 6.3 gives an L 1 L^1 L 1 ∣ D ∣ e i φ |D|e^{i\varphi} ∣ D ∣ e i φ D = − i ∂ x D=-i\partial_x D = − i ∂ x Theorem 6.4 turns it into the required bilinear energy estimate. For the ideal Poisson trace field, dividing each trace by the square root of its observation time gives a covariance error bounded by C B / s u C_B/\sqrt{su} C B / s u s , u s,u s , u
Rio’s coupling [33 ] , with its time decay retained, replaces the ideal Poisson trace field by a Gaussian field with the same covariance. For the observation-time squared process, the comparison error is O ( ( 1 + log M ) / M ) O((1+\log M)/\sqrt{M}) O (( 1 + log M ) / M ) L 1 L^1 L 1
Relation to earlier work
Grover spectra and spectral fluctuations. The determinant correspondence of Konno and Sato [21 ] , its covering decomposition [36 ] , and Grover trace formulae [20 ] provide the algebraic starting point. We use the regular-graph specialisation in [1 Corollary 3.1] , together with the old/new decomposition. The finite-size study [1 ] examined local spacings of the new Grover eigenangles after removing the inherited base spectrum. Here the observable is the full centred trace at increasing powers of the evolution operator, followed by time averaging under the unconditioned lift law. Oren and Smilansky [30 ] connect closed-walk statistics to form factors of random regular graphs, using random-matrix assumptions for higher-order predictions. Gnutzmann and Altland [11 ] study spectral correlations of quantum graphs under energy averaging. Our randomness is instead supplied by the covering permutations.
Relation to quantum graphs and periodic-orbit theory. The local Grover reflection ( 2 / d ) 11 T − I d (2/d)\mathbf{1}\mathbf{1}^{\mathsf{T}}-I_d ( 2/ d ) 1 1 T − I d U G U_G U G G G G [12 , 22 , 37 ] . Periodic-orbit expansions for quantum graphs relate form factors to closed trajectories, with the diagonal approximation producing the leading orthogonal-class ramp and correlated orbit pairs producing its first off-diagonal correction [4 , 22 ] . Trace formulae for discrete regular graphs likewise organise spectral statistics by periodic non-backtracking walks [29 ] . The question whether a spectral form factor is self-averaging goes back, in a broader semiclassical setting, to Prange [31 ] . The present fixed-base lift results concern logarithmic times, far below the Heisenberg scale. They describe both trace fluctuations and temporal averaging in a simultaneous graph-size and time limit.
Time averaging and other ensembles. Temporal self-averaging has precedents in circular unitary ensembles [15 Section VI] and, for a centred energy variance, in random surface covers [25 Theorem 1.6] . Naud [26 Theorem 1.1] and Maoz [25 Theorem 1.3 and Corollary 1.4] obtain variance limits and Gaussian limits for smoothed surface-cover statistics by taking the covering-degree limit before the smoothing limit. Rudnick and Wigman [35 Theorem 1.1] show that the smooth energy variance on a typical random hyperbolic surface approaches the GOE ensemble variance, taking the large-genus limit before the energy limits. These surface models impose relations absent from the independent edge permutations used here. Here the base graph is fixed, the edge permutations are independent, and the graph size, time and averaging width vary simultaneously.
Cipolloni, Erdős and Schröder [6 Section 1.3.1 and Theorem 2.5] study time-dependent linear statistics for Wigner and monoparametric ensembles. Our limiting covariance is described by the same leading-order orthogonal covariance form; the precise correspondence is given in § 11 . Maltsev and Osman [24 Definitions 2.2–2.4 and Theorems 2.1–2.2] prove mesoscopic variance asymptotics for two quantum-graph ensembles on polynomial mesoscopic scales. Their first ensemble combines a uniformly sampled d d d
Random lifts and cycle counts. Joint Poisson limits for fixed cycle lengths on fixed-base lifts are due to Greenhill, Janson and Ruciński [14 Lemma 5.1] . Quantitative cycle approximations and their links to polynomial traces also appear in the permutation and regular-graph models of Dumitriu, Johnson, Pal and Paquette [8 ] , Johnson and Pal [18 ] , and Johnson [17 ] . In particular, Johnson [17 Proposition 10] groups cycle indicators before applying mean-dependent Stein factors. Our estimates use classical Poisson approximation [2 , 34 ] and retains this benefit of aggregation. The fixed-base argument combines an exact conditional forcing coupling with a symmetric discrepancy estimate to obtain the explicit cutoff-dependent error in Theorem 4.1.
A close comparison is provided by Dumitriu and Zhu [9 Theorems 2.10, 4.4 and 4.7] , who use Poisson approximation of cycle counts to study global eigenvalue fluctuations in uniformly sampled biregular bipartite graphs, for fixed and growing degrees. Here the degree and the base remain fixed, while the trace power and averaging window grow with the number of sheets. Our main observable is a cumulative sum of squared traces on a single lift. Thus the cycle-count method is shared, but the ensemble, the varying observables, and the functional time-averaging limit are different.
Yu [38 Definitions 2.7–2.8 and Theorem 3.6] proves logarithmic-length Poisson approximation for cycle-count vectors in noisy fixed-base lifts, including bipartite bases. At zero noise, its cycle-count laws on simple bases agree with those considered here, with even sheet numbers in its non-bipartite definition. Its non-bipartite statement allows cutoffs D ≤ c ( H ) log n D \le c(H) \log n D ≤ c ( H ) log n c ( H ) > 0 c(H) > 0 c ( H ) > 0 n n n C B R 4 q R / N C_B R^4 q^R/N C B R 4 q R / N R ≤ ( 1 − ε ) log q N R \le(1-\varepsilon) \log_q N R ≤ ( 1 − ε ) log q N ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) § 4 identifies the limitation of the particular unweighted bound used in Yu’s proof. Yu’s model also permits noise and half-loops, which are not treated here.
Moment estimates and functional limits. The moment analysis builds on fixed-word and permutation-cycle results [23 , 27 , 32 ] , remainder estimates [10 ] , and methods for random permutation matrices [5 , 7 ] . The covariance scale can also be obtained from the general coefficient method of Hanany and Puder [16 Propositions 3.7, 5.7 and 6.8] , using their multiword coefficient estimates. We give a direct proof that retains the length dependence needed for the time-dependent observables. The empirical spectral limits of Gong, Li and Liu [13 ] concern a different object from these fluctuation laws.
The analysis of cumulative errors draws on Rio’s Poisson–normal coupling [33 Corollary 2.2] , functional Breuer–Major theory [28 ] , and its Poisson-noise counterpart [19 Theorems 1.2 and 1.5] . Theorem 10.2 handles the product-form covariance error of the normalised field instead of assuming exact stationarity. Combined with the uniform cycle and energy estimates, it yields temporal self-averaging and Brownian cumulative errors on a single lift while the graph size and observation window grow simultaneously.
Probability and path-space conventions. For probability laws on a common measurable space, let d T V ( μ , ν ) = sup A ∣ μ ( A ) − ν ( A ) ∣ d_{\mathrm{TV}}(\mu,\nu)=\sup_{A}|\mu(A)-\nu(A)| d TV ( μ , ν ) = sup A ∣ μ ( A ) − ν ( A ) ∣
W 2 ( μ , ν ) = inf { ( E [ ∥ V − W ∥ 2 2 ] ) 1 / 2 : L ( V ) = μ , L ( W ) = ν } , W_{2}(\mu,\nu)=\inf\left\{\left(\mathbb{E}[\lVert V-W\rVert_{2}^{2}]\right)^{1/2}:\mathcal{L}(V)=\mu,\ \mathcal{L}(W)=\nu\right\}, W 2 ( μ , ν ) = inf { ( E [∥ V − W ∥ 2 2 ] ) 1/2 : L ( V ) = μ , L ( W ) = ν } , where the infimum is over couplings. On a finite-dimensional Euclidean space, the bounded Lipschitz distance d B L d_{\mathrm{BL}} d BL ℓ 1 \ell^{1} ℓ 1 L ( V ) \mathcal{L}(V) L ( V ) V V V N ( m , v ) \mathcal{N}(m,v) N ( m , v ) m m m v v v
We write D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) J 1 J_{1} J 1 C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) x x x ∥ x ∥ ∞ = sup u ∈ [ 0 , 1 ] ∣ x ( u ) ∣ \lVert x\rVert_{\infty}=\sup_{u\in[0,1]}|x(u)| ∥ x ∥ ∞ = sup u ∈ [ 0 , 1 ] ∣ x ( u ) ∣ U G U_{G} U G
Organisation. Section 2 gives the spectral representation. Sections 3–4 establish the cycle approximation, and Section 5 supplies moment bounds. Section 6 proves the bilinear energy estimate. Section 7 organises the period, covariance correction and ideal field; Section 8 transfers fixed-offset limits. Section 9 gives explicit centring, a common transfer estimate and finite-size bounds. Section 10 proves the temporal limit theorems and their common-time consequences. Section 11 discusses their spectral meaning and scope.
Spectral representation and notation
We use the counting-measure inner product on ℓ 2 ( V × [ N ] ) \ell^{2}(V\times[N]) ℓ 2 ( V × [ N ])
H N n e w = { f : ∑ a = 1 N f ( v , a ) = 0 for every v ∈ V } . \mathcal{H}_{N}^{\mathrm{new}}=\left\{f:\sum_{a=1}^{N}f(v,a)=0\text{ for every }v\in V\right\}. H N new = { f : a = 1 ∑ N f ( v , a ) = 0 for every v ∈ V } . The subspace of fibre-constant functions is invariant under the symmetric adjacency matrix, and hence so is its orthogonal complement. Thus
A G N ≃ A B ⊕ A N n e w , A_{G_{N}}\simeq A_{B}\oplus A_{N}^{\mathrm{new}}, A G N ≃ A B ⊕ A N new , where A N n e w A_N^{\mathrm{new}} A N new H N n e w \mathcal{H}_N^{\mathrm{new}} H N new [1 Proposition 2.2] . The old block is deterministic, so the centred polynomial traces of the full matrix and its new restriction agree.
The Chebyshev polynomials of the first kind satisfy T 0 ( x ) = 1 T_0(x)=1 T 0 ( x ) = 1 T 1 ( x ) = x T_1(x)=x T 1 ( x ) = x T k + 1 ( x ) = 2 x T k ( x ) − T k − 1 ( x ) T_{k+1}(x)=2xT_k(x)-T_{k-1}(x) T k + 1 ( x ) = 2 x T k ( x ) − T k − 1 ( x )
Y N , k = 2 { Tr T k ( A N n e w 2 q ) − E [ Tr T k ( A N n e w 2 q ) ] } . Y_{N,k}=2\left\{\operatorname{Tr}T_k\left(\frac{A_N^{\mathrm{new}}}{2\sqrt{q}}\right)-\mathbb{E}\left[\operatorname{Tr}T_k\left(\frac{A_N^{\mathrm{new}}}{2\sqrt{q}}\right)\right]\right\}. Y N , k = 2 { Tr T k ( 2 q A N new ) − E [ Tr T k ( 2 q A N new ) ] } . The modes of orders zero, one and two are identically zero because the corresponding new-sector quantities are deterministic: dim H N n e w = n 0 ( N − 1 ) \dim\mathcal{H}_N^{\mathrm{new}}=n_0(N-1) dim H N new = n 0 ( N − 1 ) Tr A N n e w = 0 \operatorname{Tr}A_N^{\mathrm{new}}=0 Tr A N new = 0 Tr [ ( A N n e w ) 2 ] = n 0 ( N − 1 ) d \operatorname{Tr}[(A_N^{\mathrm{new}})^2]=n_0(N-1)d Tr [( A N new ) 2 ] = n 0 ( N − 1 ) d
Write
b j = Tr H B j , λ j = b j 2 j , μ N , j = E [ C N , j ] . b_j=\operatorname{Tr}H_B^j,\qquad\lambda_j=\frac{b_j}{2j},\qquad\mu_{N,j}=\mathbb{E}[C_{N,j}]. b j = Tr H B j , λ j = 2 j b j , μ N , j = E [ C N , j ] . where C N , j C_{N,j} C N , j j j j G N G_N G N b 1 = b 2 = 0 b_1=b_2=0 b 1 = b 2 = 0 Z j Z_j Z j j ≥ 3 j\ge3 j ≥ 3 λ j \lambda_j λ j
Y k = 2 q − k / 2 ∑ j ∣ k , j ≥ 3 j ( Z j − λ j ) , Y 1 = Y 2 = 0 , Y_k=2q^{-k/2}\sum_{\substack{j\mid k,\,j\ge3}}j(Z_j-\lambda_j),\qquad Y_1=Y_2=0, Y k = 2 q − k /2 j ∣ k , j ≥ 3 ∑ j ( Z j − λ j ) , Y 1 = Y 2 = 0 , Λ B ( k , ℓ ) = Cov ( Y k , Y ℓ ) = 2 q − ( k + ℓ ) / 2 ∑ j ∣ gcd ( k , ℓ ) j b j . (11) \Lambda_B(k,\ell)=\operatorname{Cov}(Y_k,Y_\ell)=2q^{-(k+\ell)/2}\sum_{j\mid\gcd(k,\ell)}jb_j. \tag*{(11)} Λ B ( k , ℓ ) = Cov ( Y k , Y ℓ ) = 2 q − ( k + ℓ ) /2 j ∣ g c d ( k , ℓ ) ∑ j b j . (11) The covariance formula follows from independence and Var ( Z j ) = λ j \operatorname{Var}(Z_j)=\lambda_j Var ( Z j ) = λ j § 4 and § 5 .
The determinant relation of [21 ] and the covering decomposition of [36 ] , in the specialisation to d d d [1 Corollary 3.1] , give, for every integer s ≥ 1 s\ge1 s ≥ 1
Tr U G N S − Tr U B S = ( N − 1 ) ( m 0 − n 0 ) ( 1 + ( − 1 ) s ) + 2 Tr T s ( A N n e w / d ) . \operatorname{Tr}U_{G_N}^{S}-\operatorname{Tr}U_B^{S}=(N-1)(m_0-n_0)(1+(-1)^s)+2\operatorname{Tr}T_s(A_N^{\mathrm{new}}/d). Tr U G N S − Tr U B S = ( N − 1 ) ( m 0 − n 0 ) ( 1 + ( − 1 ) s ) + 2 Tr T s ( A N new / d ) . The identity also holds when the lift is disconnected, by applying the determinant relation componentwise. Define the real coefficients a s , k a_{s,k} a s , k
T s ( r d x ) = ∑ k = 0 s a s , k T k ( x ) , a s , k = 0 ( k > s ) . (12) T_s(rdx)=\sum_{k=0}^{s}a_{s,k}T_k(x),\qquad a_{s,k}=0\quad(k>s). \tag*{(12)} T s ( r d x ) = k = 0 ∑ s a s , k T k ( x ) , a s , k = 0 ( k > s ) . (12) Centring this identity gives the exact formula
A N ( s ) = ∑ k = 3 s a s , k Y N , k . (13) \mathcal{A}_N(s)=\sum_{k=3}^{s}a_{s,k}Y_{N,k}. \tag*{(13)} A N ( s ) = k = 3 ∑ s a s , k Y N , k . (13) This centred identity retains all eigenvalues; only deterministic trace contributions cancel. We will compare (13) with
P s = ∑ k = 3 s a s , k Y k . (14) P_s=\sum_{k=3}^{s}a_{s,k}Y_k. \tag*{(14)} P s = k = 3 ∑ s a s , k Y k . (14) Short cycles and conditional exploration
Admissible cycles and exact means
Consider the deterministic graph on V × [ N ] V\times[N] V × [ N ] B B B admissible if, for every base edge, the pairs it requires form a partial bijection between the two incident fibres. For such a simple cycle, admissibility is equivalent to its projection being cyclically non-backtracking. Indeed, two cycle edges at a lifted vertex cannot use the same base edge, and distinct lifted vertices in a given fibre give distinct rows or columns of the corresponding partial permutation.
For a cycle α \alpha α ∣ α ∣ |\alpha| ∣ α ∣ V ( α ) V(\alpha) V ( α ) C N , j C_{N,j} C N , j j j j
I α = 1 { α ⊆ G N } , p α = P ( I α = 1 ) , C N , j = ∑ α ∈ C N , j I α . I_\alpha= 1_{\{\alpha\subseteq G_N\}}, \qquad p_\alpha= \mathbb{P}(I_\alpha=1), \qquad C_{N,j} = \sum_{\alpha\in C_{N,j}} I_\alpha. I α = 1 { α ⊆ G N } , p α = P ( I α = 1 ) , C N , j = α ∈ C N , j ∑ I α . If r e ( α ) r_e(\alpha) r e ( α ) α \alpha α e e e
p α = ∏ e ∈ E 1 ( N ) r e ( α ) , ( N ) r = N ( N − 1 ) ⋯ ( N − r + 1 ) , ( N ) 0 = 1. p_\alpha= \prod_{e\in E} \frac{1}{(N)_{r_e(\alpha)}}, \qquad(N)_r = N(N-1)\cdots(N-r+1), \qquad(N)_0=1. p α = e ∈ E ∏ ( N ) r e ( α ) 1 , ( N ) r = N ( N − 1 ) ⋯ ( N − r + 1 ) , ( N ) 0 = 1. Proposition 3.1 (Exact mean and one-sided error bound). Let W j ( B ) W_j(B) W j ( B ) j j j w ∈ W j ( B ) w\in W_j(B) w ∈ W j ( B ) a v ( w ) a_v(w) a v ( w ) v v v j j j a e ( w ) a_e(w) a e ( w ) e e e N ≥ j N\ge j N ≥ j
μ N , j = 1 2 j ∑ w ∈ W j ( B ) ∏ v ( N ) a v ( w ) ∏ e ( N ) a e ( w ) . (15) \mu_{N,j} = \frac{1}{2j}\sum_{w\in W_j(B)}\frac{\prod_v(N)_{a_v(w)}}{\prod_e(N)_{a_e(w)}}. \tag*{(15)} μ N , j = 2 j 1 w ∈ W j ( B ) ∑ ∏ e ( N ) a e ( w ) ∏ v ( N ) a v ( w ) . (15) Moreover,
0 ≤ λ j − μ N , j ≤ j ( j − 1 ) 2 N λ j . (16) 0\le\lambda_j-\mu_{N,j}\le\frac{j(j-1)}{2N}\lambda_j. \tag*{(16)} 0 ≤ λ j − μ N , j ≤ 2 N j ( j − 1 ) λ j . (16) Proof. For a fixed base walk, distinct sheet labels must be assigned to the vertex occurrences in each fibre. There are ∏ v ( N ) a v \prod_v(N)_{a_v} ∏ v ( N ) a v ∏ e ( N ) a e − 1 \prod_e(N)_{a_e}^{-1} ∏ e ( N ) a e − 1 2 j 2j 2 j (15) .
Put F ( r ) = log ( N ) r F(r)=\log(N)_r F ( r ) = log ( N ) r 0 ≤ r ≤ N 0\le r\le N 0 ≤ r ≤ N F F F F ( 0 ) = 0 F(0)=0 F ( 0 ) = 0 a v > 0 a_v>0 a v > 0
0 ≤ a e ≤ a v ( e ∋ v ) , ∑ e ∋ v a e = 2 a v . 0\le a_e\le a_v\quad(e\ni v), \qquad\sum_{e\ni v}a_e=2a_v. 0 ≤ a e ≤ a v ( e ∋ v ) , e ∋ v ∑ a e = 2 a v . Consequently, F ( a e ) ≥ ( a e / a v ) F ( a v ) F(a_e)\ge(a_e/a_v)F(a_v) F ( a e ) ≥ ( a e / a v ) F ( a v )
∑ e ∋ v F ( a e ) ≥ 2 F ( a v ) . \sum_{e\ni v}F(a_e)\ge2F(a_v). e ∋ v ∑ F ( a e ) ≥ 2 F ( a v ) . Sum over v v v ∑ e F ( a e ) ≥ ∑ v F ( a v ) \sum_eF(a_e)\ge\sum_vF(a_v) ∑ e F ( a e ) ≥ ∑ v F ( a v )
For the lower bound, use ∑ e a e = ∑ v a v = j \sum_ea_e=\sum_va_v=j ∑ e a e = ∑ v a v = j
∏ v ( N ) a v ∏ e ( N ) a e ≥ ∏ v ∏ r = 0 a v − 1 ( 1 − r N ) ≥ 1 − 1 2 N ∑ v a v ( a v − 1 ) ≥ 1 − j ( j − 1 ) 2 N . \frac{\prod_v(N)_{a_v}}{\prod_e(N)_{a_e}}
\ge\prod_v\prod_{r=0}^{a_v-1}\left(1-\frac{r}{N}\right)
\ge1-\frac{1}{2N}\sum_va_v(a_v-1)
\ge1-\frac{j(j-1)}{2N}. ∏ e ( N ) a e ∏ v ( N ) a v ≥ v ∏ r = 0 ∏ a v − 1 ( 1 − N r ) ≥ 1 − 2 N 1 v ∑ a v ( a v − 1 ) ≥ 1 − 2 N j ( j − 1 ) . Summing both bounds over the base walks, applying the normalising factor, and using ∣ W j ( B ) ∣ = b j |W_j(B)|=b_j ∣ W j ( B ) ∣ = b j (16) .
Since each row of H B H_B H B q q q b j ≤ 2 m 0 q j b_j\le2m_0q^j b j ≤ 2 m 0 q j R ≤ N R\le N R ≤ N
∑ j = 3 R j μ N , j ≤ C B q R , ∑ j = 3 R j 2 μ N , j ≤ C B R q R , ∑ j = 3 R j q j ≤ C B R q R . \sum_{j=3}^{R}j\mu_{N,j}\le C_Bq^R,\qquad
\sum_{j=3}^{R}j^2\mu_{N,j}\le C_BRq^R,\qquad
\sum_{j=3}^{R}jq^j\le C_BRq^R. j = 3 ∑ R j μ N , j ≤ C B q R , j = 3 ∑ R j 2 μ N , j ≤ C B R q R , j = 3 ∑ R j q j ≤ C B R q R . The first two estimates also hold with λ j \lambda_j λ j μ N , j \mu_{N,j} μ N , j
Forcing a cycle without destroying compatible cycles
Two admissible cycles are compatible if their combined requirements form a partial bijection over every base edge. Distinct admissible cycles need not be compatible; vertex-disjoint cycles are always compatible.
Lemma 3.2 (Forcing a partial permutation). Let π \pi π r − 1 r-1 r − 1 a a a b b b
π ′ = ( b π ( a ) ) ∘ π (17) \pi' = (b\ \pi(a)) \circ\pi \tag*{(17)} π ′ = ( b π ( a )) ∘ π (17) is uniformly distributed over the permutations satisfying all r r r π ( a ) = b \pi(a)=b π ( a ) = b
Proof. Neither b b b π ( a ) \pi(a) π ( a ) τ \tau τ r r r (17) are exactly
π = ( b c ) ∘ τ , \pi= (b\ c) \circ\tau, π = ( b c ) ∘ τ , where c c c N − r + 1 N-r+1 N − r + 1 r − 1 r-1 r − 1 π \pi π τ \tau τ τ \tau τ
Lemma 3.3 (Cycle-forcing coupling). Fix α ∈ C N , i \alpha\in\mathcal{C}_{N,i} α ∈ C N , i G N α G_N^\alpha G N α G N G_N G N
L ( G N α ) = L ( G N ∣ I α = 1 ) . (18) \mathcal{L}(G_N^\alpha)=\mathcal{L}(G_N\mid I_\alpha=1). \tag*{(18)} L ( G N α ) = L ( G N ∣ I α = 1 ) . (18) Writing J β , α = 1 { β ⊆ G N α } J_{\beta,\alpha}=\mathbf{1}\{\beta\subseteq G_N^\alpha\} J β , α = 1 { β ⊆ G N α }
J β , α ≥ I β whenever β is compatible with α . (19) J_{\beta,\alpha}\ge I_\beta\qquad\text{whenever }\beta\text{ is compatible with }\alpha. \tag*{(19)} J β , α ≥ I β whenever β is compatible with α . (19) For incompatible β \beta β J β , α = 0 J_{\beta,\alpha}=0 J β , α = 0
Proof. For each base edge, order the finitely many pairs required by α \alpha α { I α = 1 } \{I_\alpha=1\} { I α = 1 } (18) .
To prove (19) , suppose that β \beta β a ↦ b a\mapsto b a ↦ b a a a c = π − 1 ( b ) c=\pi^{-1}(b) c = π − 1 ( b ) β \beta β a a a b b b β \beta β b b b c ≠ a c\ne a c = a a ↦ b a\mapsto b a ↦ b β \beta β
Lemma 3.3 will be used separately for compatible and incompatible cycle pairs.
Lemma 3.4 (Vertex-disjoint cycles). Suppose that ∣ α ∣ = i |\alpha|=i ∣ α ∣ = i ∣ β ∣ = j |\beta|=j ∣ β ∣ = j V ( α ) ∩ V ( β ) = ∅ V(\alpha)\cap V(\beta)=\varnothing V ( α ) ∩ V ( β ) = ∅ i , j ≤ R i,j\le R i , j ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
E [ ∣ I β − J β , α ∣ ] ≤ 4 i j N p β . (20) \mathbb{E}[|I_\beta-J_{\beta,\alpha}|]\le\frac{4ij}{N}p_\beta. \tag*{(20)} E [ ∣ I β − J β , α ∣ ] ≤ N 4 ij p β . (20) Proof. Put a e = r e ( α ) a_e=r_e(\alpha) a e = r e ( α ) c e = r e ( β ) c_e=r_e(\beta) c e = r e ( β )
P ( I β = 1 ∣ I α = 1 ) p β = ∏ e ( N ) a e ( N ) c e ( N ) a e + c e . (21) \frac{\mathbb{P}(I_\beta=1\mid I_\alpha=1)}{p_\beta}
=\prod_e\frac{(N)_{a_e}(N)_{c_e}}{(N)_{a_e+c_e}}. \tag*{(21)} p β P ( I β = 1 ∣ I α = 1 ) = e ∏ ( N ) a e + c e ( N ) a e ( N ) c e . (21) Every factor is at least one. Its logarithm is
∑ r = 0 c e − 1 log ( 1 + a e N − a e − r ) ≤ a e c e N − i − j . \sum_{r=0}^{c_e-1}\log\left(1+\frac{a_e}{N-a_e-r}\right)\le\frac{a_ec_e}{N-i-j}. r = 0 ∑ c e − 1 log ( 1 + N − a e − r a e ) ≤ N − i − j a e c e . The logarithm of (21) is thus at most 2 i j / N ≤ 1 / 4 2ij/N\le1/4 2 ij / N ≤ 1/4 e x − 1 ≤ 2 x e^x-1\le2x e x − 1 ≤ 2 x (19) gives (20) .
Cycles intersecting a forced cycle
Deferred decisions. Fix the deterministic input of an exploration algorithm whose next query and stopping decision depend only on the answers received so far. Queries may evaluate a permutation or its inverse. For a transcript h h h F e ( h ) F_e(h) F e ( h ) e e e
{ transcript = h } = ⋂ e ∈ E { σ e ⊇ F e ( h ) } . \{\text{transcript}=h\}=\bigcap_{e\in E}\{\sigma_e\supseteq F_e(h)\}. { transcript = h } = e ∈ E ⋂ { σ e ⊇ F e ( h )} . Induction over the successive queries shows that every completion on the right reproduces the transcript. Thus, conditional on h h h § 5 .
Proposition 3.5 (Conditional overlap bound). Fix an admissible cycle α \alpha α i i i N > i + j N>i+j N > i + j j ≥ 3 j\ge3 j ≥ 3
E [ ∑ β ∈ C N , j , β ≠ α V ( β ) ∩ V ( α ) ≠ ∅ I β | I α = 1 ] ≤ i 2 j d q j − 1 N − i − j . (22) \mathbb{E}\left[\sum_{\substack{\beta\in\mathcal{C}_{N,j},\ \beta\ne\alpha\\ V(\beta)\cap V(\alpha)\ne\varnothing}} I_\beta\ \middle|\ I_\alpha=1\right]\le\frac{i^2j\,d q^{j-1}}{N-i-j}. \tag*{(22)} E β ∈ C N , j , β = α V ( β ) ∩ V ( α ) = ∅ ∑ I β I α = 1 ≤ N − i − j i 2 j d q j − 1 . (22) In particular, if i , j ≤ R i,j\le R i , j ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 C B i 2 j q j / N C_B i^2j q^j/N C B i 2 j q j / N
Proof. Expose all pairs belonging to α \alpha α
Choose a starting vertex x ∈ V ( α ) x\in V(\alpha) x ∈ V ( α ) j j j x x x
Throughout this exploration, at most i + j i+j i + j V ( α ) V(\alpha) V ( α )
i N − i − j . \frac{i}{N-i-j}. N − i − j i . Define the event at a step to be empty when no fresh pair is exposed. Summing the conditional probabilities over at most j j j j i / ( N − i − j ) ji/(N-i-j) j i / ( N − i − j )
If the lifted trajectory is a simple cycle β ≠ α \beta\ne\alpha β = α α \alpha α V ( α ) V(\alpha) V ( α ) α \alpha α α \alpha α α \alpha α V ( α ) V(\alpha) V ( α ) α \alpha α β \beta β V ( α ) V(\alpha) V ( α )
There are i i i x x x d q j − 1 dq^{j-1} d q j − 1 (22) gives at least one such rooted trajectory (in fact 2 ∣ V ( α ) ∩ V ( β ) ∣ 2|V(\alpha)\cap V(\beta)| 2∣ V ( α ) ∩ V ( β ) ∣
i d q j − 1 j i N − i − j . i\,dq^{j-1}\frac{ji}{N-i-j}. i d q j − 1 N − i − j j i . Lemma 3.6 (Unconditional overlap bound). For a fixed set A ⊂ V × [ N ] A\subset V\times[N] A ⊂ V × [ N ] i i i
E [ ∑ β ∈ C N , j V ( β ) ∩ A ≠ ∅ I β ] ≤ j i N μ N , j . (23) \mathbb{E}\left[\sum_{\substack{\beta\in\mathcal{C}_{N,j}\\V(\beta)\cap A\ne\varnothing}} I_\beta\right]\leq\frac{ji}{N}\mu_{N,j}. \tag*{(23)} E β ∈ C N , j V ( β ) ∩ A = ∅ ∑ I β ≤ N j i μ N , j . (23) Proof. Let T N , j ( x ) T_{N,j}(x) T N , j ( x ) j j j x x x E [ T N , j ( x ) ] \mathbb{E}[T_{N,j}(x)] E [ T N , j ( x )] x x x v v v
E [ T N , j ( x ) ] = 1 N E [ ∑ y ∈ { v } × [ N ] T N , j ( y ) ] ≤ j N μ N , j . \mathbb{E}[T_{N,j}(x)]=\frac{1}{N}\mathbb{E}\left[\sum_{y\in\{v\}\times[N]}T_{N,j}(y)\right]\leq\frac{j}{N}\mu_{N,j}. E [ T N , j ( x )] = N 1 E y ∈ { v } × [ N ] ∑ T N , j ( y ) ≤ N j μ N , j . Summing over x ∈ A x\in A x ∈ A (23) .
Length-aggregated Poisson approximation
We use the classical immigration–death construction for the Poisson Stein equation [2 , 8 ] , together with a coordinatewise bound in terms of the aggregated means. The second-difference estimate below includes diagonal coordinate pairs.
Theorem 4.1 (Uniform cycle approximation). There is a constant C B < ∞ C_B<\infty C B < ∞ R ≥ 3 R\geq3 R ≥ 3 N ≥ 8 R 2 N\geq8R^2 N ≥ 8 R 2
d T V ( L ( C N , 3 , … , C N , R ) , ⨂ j = 3 R Poisson ( λ j ) ) ≤ C B R 4 q R N . (24) d_{\mathrm{TV}}\left(\mathcal{L}(C_{N,3},\ldots,C_{N,R}),\bigotimes_{j=3}^{R}\operatorname{Poisson}(\lambda_j)\right)\leq C_B\frac{R^4q^R}{N}. \tag*{(24)} d TV ( L ( C N , 3 , … , C N , R ) , j = 3 ⨂ R Poisson ( λ j ) ) ≤ C B N R 4 q R . (24) Primitivity of H B H_B H B
Mean-sensitive second differences
Let Z Z Z Z i ∼ Poisson ( μ i ) Z_i\sim\operatorname{Poisson}(\mu_i) Z i ∼ Poisson ( μ i ) i i i J J J μ i > 0 \mu_i>0 μ i > 0 Z ≥ 0 J \mathbb{Z}_{\geq0}^{J} Z ≥ 0 J e i e_i e i i i i Δ i h ( x ) = h ( x + e i ) − h ( x ) \Delta_i h(x)=h(x+e_i)-h(x) Δ i h ( x ) = h ( x + e i ) − h ( x ) x ∈ Z ≥ 0 J x\in\mathbb{Z}_{\geq0}^{J} x ∈ Z ≥ 0 J
A h ( x ) = ∑ i μ i Δ i h ( x ) + ∑ i x i { h ( x − e i ) − h ( x ) } . \mathcal{A}h(x)=\sum_i\mu_i\Delta_i h(x)+\sum_i x_i\{h(x-e_i)-h(x)\}. A h ( x ) = i ∑ μ i Δ i h ( x ) + i ∑ x i { h ( x − e i ) − h ( x )} . Terms with x i = 0 x_i=0 x i = 0 f f f A h = f − E [ f ( Z ) ] \mathcal{A}h=f-\mathbb{E}[f(Z)] A h = f − E [ f ( Z )]
h ( x ) = − ∫ 0 ∞ { E [ f ( X t x ) ] − E [ f ( Z ) ] } d t . (25) h(x)=-\int_0^\infty\{\mathbb{E}[f(X_t^x)]-\mathbb{E}[f(Z)]\}\,\mathrm{d}t. \tag*{(25)} h ( x ) = − ∫ 0 ∞ { E [ f ( X t x )] − E [ f ( Z )]} d t . (25) where X t x X_t^x X t x X 0 x = x X_0^x=x X 0 x = x μ i \mu_i μ i ( ∥ x ∥ 1 + ∑ i μ i ) e − t (\lVert x\rVert_1+\sum_i\mu_i)e^{-t} (∥ x ∥ 1 + ∑ i μ i ) e − t
Write log + a = max ( 0 , log a ) \log^+a=\max(0,\log a) log + a = max ( 0 , log a ) a > 0 a>0 a > 0
ρ ( a ) = min { 1 , 2 + log + a a } . \rho(a)=\min\left\{1,\frac{2+\log^+a}{a}\right\}. ρ ( a ) = min { 1 , a 2 + log + a } . Lemma 4.2 (Coordinatewise second differences). For the solution (4.2) and every pair i , j i,j i , j including i = j i=j i = j
∥ Δ i Δ j h ∥ ∞ ≤ ρ ( μ i μ j ) . (26) \lVert\Delta_i\Delta_jh\rVert_\infty\leq\rho\left(\sqrt{\mu_i\mu_j}\right). \tag*{(26)} ∥ Δ i Δ j h ∥ ∞ ≤ ρ ( μ i μ j ) . (26) The bound is independent of ∣ J ∣ \lvert J\rvert ∣ J ∣ and of x x x
Proof. Couple chains started at x x x x + e i x+e_i x + e i x + e j x+e_j x + e j x + e i + e j x+e_i+e_j x + e i + e j 2 e − 2 t 2e^{-2t} 2 e − 2 t
Δ i Δ j h ( x ) = − ∫ 0 ∞ e − 2 t E [ Δ i Δ j f ( X t x ) ] d t . \Delta_i\Delta_jh(x)=-\int_0^\infty e^{-2t}\mathbb{E}\left[\Delta_i\Delta_jf\left(X_t^x\right)\right]\,dt. Δ i Δ j h ( x ) = − ∫ 0 ∞ e − 2 t E [ Δ i Δ j f ( X t x ) ] d t . At time t t t μ i u \mu_i u μ i u u = 1 − e − t u=1-e^{-t} u = 1 − e − t
For v > 0 v>0 v > 0 p v p_v p v Z ≥ 0 \mathbb{Z}_{\geq0} Z ≥ 0 δ p v ( n ) = p v ( n − 1 ) − p v ( n ) \delta p_v(n)=p_v(n-1)-p_v(n) δ p v ( n ) = p v ( n − 1 ) − p v ( n ) p v ( − 1 ) = 0 p_v(-1)=0 p v ( − 1 ) = 0
∥ δ p v ∥ ℓ 1 = E [ ∣ Z v v − 1 ∣ ] ≤ v − 1 / 2 , Z v ∼ Poisson ( v ) . (27) \lVert\delta p_v\rVert_{\ell^1}=\mathbb{E}\left[\left|\frac{Z_v}{v}-1\right|\right]\leq v^{-1/2},\qquad Z_v\sim\operatorname{Poisson}(v). \tag*{(27)} ∥ δ p v ∥ ℓ 1 = E [ v Z v − 1 ] ≤ v − 1/2 , Z v ∼ Poisson ( v ) . (27) For i ≠ j i\ne j i = j ℓ 1 \ell^1 ℓ 1
∣ E [ Δ i Δ j f ( X t x ) ] ∣ ≤ 1 2 u μ i μ j . \left|\mathbb{E}\left[\Delta_i\Delta_jf\left(X_t^x\right)\right]\right|\leq\frac{1}{2u\sqrt{\mu_i\mu_j}}. ∣ E [ Δ i Δ j f ( X t x ) ] ∣ ≤ 2 u μ i μ j 1 . For i = j i=j i = j μ i u \mu_i u μ i u μ i u / 2 \mu_i u/2 μ i u /2 δ p μ i u / 2 ∗ δ p μ i u / 2 \delta p_{\mu_i u/2}*\delta p_{\mu_i u/2} δ p μ i u /2 ∗ δ p μ i u /2 ℓ 1 \ell^1 ℓ 1 2 / ( μ i u ) 2/(\mu_i u) 2/ ( μ i u ) 1 / ( μ i u ) 1/(\mu_i u) 1/ ( μ i u )
In both cases, the elementary bound ∣ Δ i Δ j f ∣ ≤ 2 \lvert\Delta_i\Delta_jf\rvert\leq2 ∣ Δ i Δ j f ∣ ≤ 2 a = μ i μ j a=\sqrt{\mu_i\mu_j} a = μ i μ j
∥ Δ i Δ j h ∥ ∞ ≤ ∫ 0 ∞ e − 2 t min { 2 , 1 a ( 1 − e − t ) } d t = ∫ 0 1 ( 1 − u ) min { 2 , 1 a u } d u = { 1 , a ≤ 1 / 2 , ( log ( 2 a ) ) / a + 1 / ( 4 a 2 ) , a > 1 / 2. \begin{aligned}
\lVert\Delta_i\Delta_jh\rVert_\infty\leq\int_0^\infty e^{-2t}\min\left\{2,\frac{1}{a(1-e^{-t})}\right\}\,dt \\
&=\int_0^1(1-u)\min\left\{2,\frac{1}{au}\right\}\,du
=\begin{cases}
1, & a\leq1/2,\\
(\log(2a))/a+1/(4a^2), & a>1/2.
\end{cases}
\end{aligned} ∥ Δ i Δ j h ∥ ∞ ≤ ∫ 0 ∞ e − 2 t min { 2 , a ( 1 − e − t ) 1 } d t = ∫ 0 1 ( 1 − u ) min { 2 , a u 1 } d u = { 1 , ( log ( 2 a )) / a + 1/ ( 4 a 2 ) , a ≤ 1/2 , a > 1/2. This is at most one, and at most ( 2 + log + a ) / a (2+\log^+a)/a ( 2 + log + a ) / a (26) follows. All intermediate bounds are uniform over the conditioned values of the survivors, so the bound is uniform in the number of coordinates. □ \square □
Logarithmic dependence on the means is already present in [2 Lemma 3] , and [34 Section 3] discusses why logarithmic factors in multivariate Stein estimates cannot be discarded without further structure. We make no claim that the polynomial loss in our graph application is optimal.
Aggregation before estimation
Lemma 4.3 (Aggregation before estimation). Let Γ = ⨆ i ∈ J Γ i \Gamma= \bigsqcup_{i\in J}\Gamma_i Γ = ⨆ i ∈ J Γ i I = ( I α ) α ∈ Γ I=(I_\alpha)_{\alpha\in\Gamma} I = ( I α ) α ∈ Γ p α = P ( I α = 1 ) > 0 p_\alpha=\mathbb{P}(I_\alpha=1)>0 p α = P ( I α = 1 ) > 0 C i = ∑ α ∈ Γ i I α C_i=\sum_{\alpha\in\Gamma_i}I_\alpha C i = ∑ α ∈ Γ i I α C = ( C i ) i ∈ J C=(C_i)_{i\in J} C = ( C i ) i ∈ J μ i = E [ C i ] > 0 \mu_i=\mathbb{E}[C_i]>0 μ i = E [ C i ] > 0 α \alpha α J α = ( J β , α ) β J^\alpha=(J_{\beta,\alpha})_\beta J α = ( J β , α ) β I I I I α = 1 I_\alpha=1 I α = 1
S i j = ∑ α ∈ Γ i p α ∑ β ∈ Γ j β ≠ α E [ ∣ I β − J β , α ∣ ] , M i j = ρ ( μ i μ j ) . S_{ij}=\sum_{\alpha\in\Gamma_i}p_\alpha\sum_{\substack{\beta\in\Gamma_j\\\beta\ne\alpha}}\mathbb{E}[|I_\beta-J_{\beta,\alpha}|],\qquad M_{ij}=\rho(\sqrt{\mu_i\mu_j}). S ij = α ∈ Γ i ∑ p α β ∈ Γ j β = α ∑ E [ ∣ I β − J β , α ∣ ] , M ij = ρ ( μ i μ j ) . Then
d T V ( L ( C ) , ⨂ i Poisson ( μ i ) ) ≤ ∑ i M i i ∑ α ∈ Γ i p α 2 + ∑ i , j M i j S i j . (28) d_{\mathrm{TV}}\left(\mathcal{L}(C),\bigotimes_i\operatorname{Poisson}(\mu_i)\right)\le\sum_i M_{ii}\sum_{\alpha\in\Gamma_i}p_\alpha^2+\sum_{i,j}M_{ij}S_{ij}. \tag*{(28)} d TV ( L ( C ) , i ⨂ Poisson ( μ i ) ) ≤ i ∑ M ii α ∈ Γ i ∑ p α 2 + i , j ∑ M ij S ij . (28) Proof. Set C j α = ∑ β ∈ Γ j J β , α C_j^\alpha=\sum_{\beta\in\Gamma_j}J_{\beta,\alpha} C j α = ∑ β ∈ Γ j J β , α α ∈ Γ i \alpha\in\Gamma_i α ∈ Γ i C i α C_i^\alpha C i α
E [ A h ( C ) ] = ∑ i ∑ α ∈ Γ i p α E [ Δ i h ( C ) − Δ i h ( C α − e i ) ] . \mathbb{E}[Ah(C)]=\sum_i\sum_{\alpha\in\Gamma_i}p_\alpha\mathbb{E}[\Delta_i h(C)-\Delta_i h(C^\alpha-e_i)]. E [ A h ( C )] = i ∑ α ∈ Γ i ∑ p α E [ Δ i h ( C ) − Δ i h ( C α − e i )] . Telescoping between the two nonnegative lattice points and applying Lemma 4.2 yields
∣ Δ i h ( x ) − Δ i h ( y ) ∣ ≤ ∑ j M i j ∣ x j − y j ∣ . |\Delta_i h(x)-\Delta_i h(y)|\le\sum_j M_{ij}|x_j-y_j|. ∣ Δ i h ( x ) − Δ i h ( y ) ∣ ≤ j ∑ M ij ∣ x j − y j ∣. For j ≠ i j\ne i j = i ∣ I β − J β , α ∣ |I_\beta-J_{\beta,\alpha}| ∣ I β − J β , α ∣ i i i J α , α = 1 J_{\alpha,\alpha}=1 J α , α = 1
∣ C i − ( C i α − 1 ) ∣ ≤ I α + ∑ β ∈ Γ i β ≠ α ∣ I β − J β , α ∣ . |C_i-(C_i^\alpha-1)|\le I_\alpha+\sum_{\substack{\beta\in\Gamma_i\\\beta\ne\alpha}}|I_\beta-J_{\beta,\alpha}|. ∣ C i − ( C i α − 1 ) ∣ ≤ I α + β ∈ Γ i β = α ∑ ∣ I β − J β , α ∣. Taking expectations gives the self term p α 2 p_\alpha^2 p α 2 S i j S_{ij} S ij M i j M_{ij} M ij μ i \mu_i μ i p α p_\alpha p α
Symmetric cycle discrepancies
For the remainder of this section, write Γ i = C N , i \Gamma_i=C_{N,i} Γ i = C N , i C i = C N , i C_i=C_{N,i} C i = C N , i μ i = μ N , i \mu_i=\mu_{N,i} μ i = μ N , i Lemma 3.3 and let S i j S_{ij} S ij Lemma 4.3 .
Lemma 4.4 (Pairwise symmetry). For distinct admissible cycles,
p α E [ ∣ I β − J β , α ∣ ] = ∣ Cov ( I α , I β ) ∣ . (29) p_\alpha\mathbb{E}[|I_\beta-J_{\beta,\alpha}|]=|\operatorname{Cov}(I_\alpha,I_\beta)|. \tag*{(29)} p α E [ ∣ I β − J β , α ∣ ] = ∣ Cov ( I α , I β ) ∣. (29) Consequently S i j = S j i S_{ij}=S_{ji} S ij = S j i
Proof. For compatible cycles, preservation in Lemma 3.3 gives
p α E [ ∣ I β − J β , α ∣ ] = p α { P ( I β = 1 ∣ I α = 1 ) − p β } = Cov ( I α , I β ) ≥ 0. p_\alpha\mathbb{E}[|I_\beta-J_{\beta,\alpha}|]=p_\alpha\{\mathbb{P}(I_\beta=1\mid I_\alpha=1)-p_\beta\}=\operatorname{Cov}(I_\alpha,I_\beta)\ge0. p α E [ ∣ I β − J β , α ∣ ] = p α { P ( I β = 1 ∣ I α = 1 ) − p β } = Cov ( I α , I β ) ≥ 0. For incompatible cycles, J β , α = 0 J_{\beta,\alpha}=0 J β , α = 0 p α p β p_\alpha p_\beta p α p β i = j i=j i = j
Lemma 4.5 (Bidirectional discrepancy estimate). If i , j ≤ R i,j\le R i , j ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 μ i μ j > 0 \mu_i\mu_j>0 μ i μ j > 0
S i j μ i μ j ≤ ( 5 m 0 + 2 d q ) ( i j ) 3 / 2 q ( i + j ) / 2 N . (30) \frac{S_{ij}}{\sqrt{\mu_i\mu_j}}\le\left(5m_0+\frac{2d}{q}\right)\frac{(ij)^{3/2}q^{(i+j)/2}}{N}. \tag*{(30)} μ i μ j S ij ≤ ( 5 m 0 + q 2 d ) N ( ij ) 3/2 q ( i + j ) /2 . (30) Proof. Lemma 3.4 bounds the vertex-disjoint contribution by 4 i j μ i μ j / N 4ij\mu_i\mu_j/N 4 ij μ i μ j / N
E [ ∣ I β − J β , α ∣ ] ≤ p β + P ( I β = 1 ∣ I α = 1 ) . \mathbb{E}[|I_\beta-J_{\beta,\alpha}|]\le p_\beta+\mathbb{P}(I_\beta=1\mid I_\alpha=1). E [ ∣ I β − J β , α ∣ ] ≤ p β + P ( I β = 1 ∣ I α = 1 ) . Lemma 3.6 bounds the sum of the unconditional terms. Proposition 3.5, with N − i − j ≥ N / 2 N-i-j\ge N/2 N − i − j ≥ N /2
S i j ≤ 5 i j μ i μ j N + 2 d q i 2 j μ i q j N . (31) S_{ij}\le\frac{5ij\mu_i\mu_j}{N}+\frac{2d}{q}\frac{i^2j\mu_iq^j}{N}. \tag*{(31)} S ij ≤ N 5 ij μ i μ j + q 2 d N i 2 j μ i q j . (31) By Lemma 4.4, (31) also holds with i , j i,j i , j
min { i 2 j μ i q j , j 2 i μ j q i } ≤ ( i j ) 3 / 2 μ i μ j q ( i + j ) / 2 . \min\{i^2j\mu_iq^j,j^2i\mu_jq^i\}\le(ij)^{3/2}\sqrt{\mu_i\mu_j}q^{(i+j)/2}. min { i 2 j μ i q j , j 2 i μ j q i } ≤ ( ij ) 3/2 μ i μ j q ( i + j ) /2 . Proposition 3.1 and b i ≤ 2 m 0 q i b_i\le2m_0q^i b i ≤ 2 m 0 q i μ i ≤ m 0 q i / i \mu_i\le m_0q^i/i μ i ≤ m 0 q i / i μ i μ j \sqrt{\mu_i\mu_j} μ i μ j 5 m 0 i j q ( i + j ) / 2 5m_0\sqrt{ij}\,q^{(i+j)/2} 5 m 0 ij q ( i + j ) /2 i j \sqrt{ij} ij ( i j ) 3 / 2 (ij)^{3/2} ( ij ) 3/2 (30) . No uniform positive lower bound on the nonzero aggregated means is needed. □ \square □
Proof of Theorem 4.1. If μ i = 0 \mu_i=0 μ i = 0 C i = 0 C_i=0 C i = 0 (16) and N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 μ i ≥ ( 15 / 16 ) λ i \mu_i\ge(15/16)\lambda_i μ i ≥ ( 15/16 ) λ i λ i = 0 \lambda_i=0 λ i = 0 J J J J J J
Set
L R = 2 + log m 0 + R log q . L_R=2+\log m_0+R\log q. L R = 2 + log m 0 + R log q . Since μ i ≤ m 0 q R \mu_i\le m_0q^R μ i ≤ m 0 q R
M i j ≤ L R μ i μ j , L R ≤ C B R . (32) M_{ij}\le\frac{L_R}{\sqrt{\mu_i\mu_j}},\qquad L_R\le C_BR. \tag*{(32)} M ij ≤ μ i μ j L R , L R ≤ C B R . (32) Apply Lemma 4.3 with the exact forcing couplings. For α ∈ Γ i \alpha\in\Gamma_i α ∈ Γ i p α ≤ ( N − R ) − i ≤ ( N − R ) − 3 p_\alpha\le(N-R)^{-i}\le(N-R)^{-3} p α ≤ ( N − R ) − i ≤ ( N − R ) − 3
∑ i ∈ J M i i ∑ α ∈ Γ i p α 2 ≤ L R ∣ J ∣ ( N − R ) 3 ≤ L R R ( N − R ) 3 . (33) \sum_{i\in J}M_{ii}\sum_{\alpha\in\Gamma_i}p_\alpha^2
\le\frac{L_R|J|}{(N-R)^3}
\le\frac{L_RR}{(N-R)^3}. \tag*{(33)} i ∈ J ∑ M ii α ∈ Γ i ∑ p α 2 ≤ ( N − R ) 3 L R ∣ J ∣ ≤ ( N − R ) 3 L R R . (33) For the distinct-pair term, Lemma 4.5 and (32) give
∑ i , j ∈ J M i j S i j ≤ L R ( 5 m 0 + 2 d / q ) N ( ∑ i = 3 R i 3 / 2 q i / 2 ) 2 ≤ L R ( 5 m 0 + 2 d / q ) R 3 q R ( 1 − q − 1 / 2 ) 2 N ≤ C B R 4 q R N . \begin{aligned}
\sum_{i,j\in J}M_{ij}S_{ij}
\le\frac{L_R(5m_0+2d/q)}{N}
\left(\sum_{i=3}^{R}i^{3/2}q^{i/2}\right)^2 \\
&\le\frac{L_R(5m_0+2d/q)R^3q^R}{(1-q^{-1/2})^2N}
\le C_B\frac{R^4q^R}{N}.
\end{aligned} i , j ∈ J ∑ M ij S ij ≤ N L R ( 5 m 0 + 2 d / q ) ( i = 3 ∑ R i 3/2 q i /2 ) 2 ≤ ( 1 − q − 1/2 ) 2 N L R ( 5 m 0 + 2 d / q ) R 3 q R ≤ C B N R 4 q R . We have thus compared the vector of cycle counts with independent Poisson variables of means μ i \mu_i μ i λ i \lambda_i λ i λ i − μ i \lambda_i-\mu_i λ i − μ i
∑ i = 3 R ( λ i − μ i ) ≤ m 0 2 N ∑ i = 3 R ( i − 1 ) q i ≤ m 0 R q R 2 ( 1 − q − 1 ) N . (34) \sum_{i=3}^{R}(\lambda_i-\mu_i)
\le\frac{m_0}{2N}\sum_{i=3}^{R}(i-1)q^i
\le\frac{m_0Rq^R}{2(1-q^{-1})N}. \tag*{(34)} i = 3 ∑ R ( λ i − μ i ) ≤ 2 N m 0 i = 3 ∑ R ( i − 1 ) q i ≤ 2 ( 1 − q − 1 ) N m 0 R q R . (34) Combining (33) –(34) proves (24) . In particular, an explicit, unoptimised upper bound is
L R R ( N − R ) 3 + L R ( 5 m 0 + 2 d / q ) R 3 q R ( 1 − q − 1 / 2 ) 2 N + m 0 R q R 2 ( 1 − q − 1 ) N . \frac{L_RR}{(N-R)^3}
+\frac{L_R(5m_0+2d/q)R^3q^R}{(1-q^{-1/2})^2N}
+\frac{m_0Rq^R}{2(1-q^{-1})N}. ( N − R ) 3 L R R + ( 1 − q − 1/2 ) 2 N L R ( 5 m 0 + 2 d / q ) R 3 q R + 2 ( 1 − q − 1 ) N m 0 R q R . All denominators are positive under the stated hypothesis. □ \square □
Comparison with an unweighted neighbourhood estimate. We compare with [38 Theorem 6.1 and Section 6, p. 28] at zero noise on a simple non-bipartite base. Its parameters are H = B H=B H = B m = N m=N m = N n = n 0 N n=n_0N n = n 0 N D = R D=R D = R N N N S α S_\alpha S α 2 ∣ α ∣ 2|\alpha| 2∣ α ∣ n 0 N d n_0Nd n 0 N d β ∈ N α \beta\in\mathcal{N}_\alpha β ∈ N α S α ∩ S β ≠ ∅ S_\alpha\cap S_\beta\ne\varnothing S α ∩ S β = ∅ [38 Theorem 6.1] by
b 1 = ∑ α ∑ β ∈ N α p α p β . b_1=\sum_\alpha\sum_{\beta\in\mathcal{N}_\alpha}p_\alpha p_\beta. b 1 = α ∑ β ∈ N α ∑ p α p β . Retain only length-R R R u x = ∑ α ∈ C N , R : x ∈ S α p α u_x=\sum_{\alpha\in C_{N,R}:x\in S_\alpha}p_\alpha u x = ∑ α ∈ C N , R : x ∈ S α p α
∑ x u x = 2 R μ N , R , 2 R b 1 ≥ ∑ x u x 2 ≥ ( 2 R μ N , R ) 2 n 0 N d . \sum_x u_x=2R\mu_{N,R},\qquad2Rb_1\ge\sum_x u_x^2\ge\frac{(2R\mu_{N,R})^2}{n_0Nd}. x ∑ u x = 2 R μ N , R , 2 R b 1 ≥ x ∑ u x 2 ≥ n 0 N d ( 2 R μ N , R ) 2 . The first inequality uses ∣ S α ∩ S β ∣ ≤ 2 R |S_\alpha\cap S_\beta|\le2R ∣ S α ∩ S β ∣ ≤ 2 R H B H_B H B R → ∞ R\to\infty R → ∞ R 2 / N → 0 R^2/N\to0 R 2 / N → 0 Proposition 3.1 and b R / q R → 1 b_R/q^R\to1 b R / q R → 1
μ N , R ∼ q R 2 R , b 1 ≥ c B q 2 R N R \mu_{N,R}\sim\frac{q^R}{2R},\qquad b_1\ge c_B\frac{q^{2R}}{NR} μ N , R ∼ 2 R q R , b 1 ≥ c B N R q 2 R for some c B > 0 c_B>0 c B > 0 N N N R = ⌊ ξ log q N ⌋ R=\lfloor\xi\log_q N\rfloor R = ⌊ ξ log q N ⌋ 1 / 2 < ξ < 1 1/2<\xi<1 1/2 < ξ < 1 R 4 q R / N → 0 R^4q^R/N\to0 R 4 q R / N → 0 2 ( 2 b 1 + 2 b 2 + b 3 ) 2(2b_1+2b_2+b_3) 2 ( 2 b 1 + 2 b 2 + b 3 ) [38 Theorem 6.1] is not informative in this range with those neighbourhoods. The nonnegative quantities b 2 , b 3 b_2,b_3 b 2 , b 3
Non-backtracking traces and moment estimates
The non-cycle remainder
Put W N , k = Tr H G N k W_{N,k}=\operatorname{Tr}H_{G_N}^k W N , k = Tr H G N k j j j 2 j 2j 2 j k k k j ∣ k j\mid k j ∣ k
W N , k = ∑ j ∣ k j ≥ 3 2 j C N , j + R N , k , R N , k ≥ 0. (35) W_{N,k}=\sum_{\substack{j\mid k\\j\ge3}}2jC_{N,j}+R_{N,k},\qquad R_{N,k}\ge0. \tag*{(35)} W N , k = j ∣ k j ≥ 3 ∑ 2 j C N , j + R N , k , R N , k ≥ 0. (35) where R N , k R_{N,k} R N , k [10 Lemma 1.3.5 and the following paragraph] . We include an exploration proof, since the same construction is used to estimate pairs of walks.
Lemma 5.1 (Two-collision bound). For N ≥ 2 k N\ge2k N ≥ 2 k k ≥ 3 k\ge3 k ≥ 3
E [ R N , k ] ≤ C B k 4 q k N . \mathbb{E}[R_{N,k}]\le C_B\frac{k^4q^k}{N}. E [ R N , k ] ≤ C B N k 4 q k . Proof. Fix a lifted starting vertex and a prescribed non-backtracking base walk of length k k k collision is a newly exposed edge whose endpoint is a previously visited vertex. At each step, conditional on the complete exploration history, its probability is at most k / ( N − k ) k/(N-k) k / ( N − k )
If S S S ∣ E ( S ) ∣ − ∣ V ( S ) ∣ + 1 |E(S)|-|V(S)|+1 ∣ E ( S ) ∣ − ∣ V ( S ) ∣ + 1 S S S ∣ E ( S ) ∣ = ∣ V ( S ) ∣ |E(S)|=|V(S)| ∣ E ( S ) ∣ = ∣ V ( S ) ∣ S S S R N , k R_{N,k} R N , k
For any two step indices, conditioning at the later step and then at the earlier step bounds the probability of two collisions by ( k / ( N − k ) ) 2 (k/(N-k))^2 ( k / ( N − k ) ) 2 n 0 N n_0N n 0 N d q k − 1 dq^{k-1} d q k − 1 k k k
E [ R N , k ] ≤ n 0 N d q k − 1 ( k 2 ) ( k N − k ) 2 ≤ C B k 4 q k N . \mathbb{E}[R_{N,k}] \le n_0Ndq^{k-1}\binom{k}{2}\left(\frac{k}{N-k}\right)^2 \le C_B\frac{k^4q^k}{N}. E [ R N , k ] ≤ n 0 N d q k − 1 ( 2 k ) ( N − k k ) 2 ≤ C B N k 4 q k . Including trajectories that are not closed only enlarges the upper bound.
The classical non-backtracking spectral identity for a finite d d d
Tr H G k = 2 q k / 2 Tr T k ( A G 2 q ) + ( ∣ E ( G ) ∣ − ∣ V ( G ) ∣ ) ( 1 + ( − 1 ) k ) . (36) \operatorname{Tr} H_G^k=2q^{k/2}\operatorname{Tr}T_k\left(\frac{A_G}{2\sqrt{q}}\right)+\left(|E(G)|-|V(G)|\right)\left(1+(-1)^k\right). \tag*{(36)} Tr H G k = 2 q k /2 Tr T k ( 2 q A G ) + ( ∣ E ( G ) ∣ − ∣ V ( G ) ∣ ) ( 1 + ( − 1 ) k ) . (36) For example, it follows from the non-backtracking eigenvalue description in [14 Lemma 5.3] : each adjacency eigenvalue a a a z 2 − a z + q = 0 z^2-az+q=0 z 2 − a z + q = 0 k k k 2 q k / 2 T k ( a / ( 2 q ) ) 2q^{k/2}T_k(a/(2\sqrt{q})) 2 q k /2 T k ( a / ( 2 q )) 1 1 1 − 1 -1 − 1 ∣ E ( G ) ∣ − ∣ V ( G ) ∣ |E(G)|-|V(G)| ∣ E ( G ) ∣ − ∣ V ( G ) ∣
Since the old block in § 2 and the last term of (5.2) are deterministic,
Y N , k = q − k / 2 ( W N , k − E [ W N , k ] ) . (37) Y_{N,k}=q^{-k/2}(W_{N,k}-\mathbb{E}[W_{N,k}]). \tag*{(37)} Y N , k = q − k /2 ( W N , k − E [ W N , k ]) . (37) Combining (5.1) and (5.3) gives
Y N , k = 2 q − k / 2 ∑ j ∣ k j ≥ 3 j ( C N , j − μ N , j ) + q − k / 2 ( R N , k − E [ R N , k ] ) . (38) Y_{N,k}=2q^{-k/2}\sum_{\substack{j\mid k\\j\ge3}}j(C_{N,j}-\mu_{N,j})+q^{-k/2}(R_{N,k}-\mathbb{E}[R_{N,k}]). \tag*{(38)} Y N , k = 2 q − k /2 j ∣ k j ≥ 3 ∑ j ( C N , j − μ N , j ) + q − k /2 ( R N , k − E [ R N , k ]) . (38) The last term has expected absolute value at most C B k 4 q k / 2 / N C_Bk^4q^{k/2}/N C B k 4 q k /2 / N N N N
Cycle-count covariances
We estimate the second moments directly using the forcing couplings.
Proposition 5.2 (Cycle covariance error). For 3 ≤ i , j ≤ R 3\le i,j\le R 3 ≤ i , j ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
∣ Cov ( C N , i , C N , j ) − 1 { i = j } λ i ∣ ≤ C B i j q i + j N . (39) \left|\operatorname{Cov}(C_{N,i},C_{N,j})-\mathbf{1}_{\{i=j\}}\lambda_i\right|\le C_B\frac{ijq^{i+j}}{N}. \tag*{(39)} Cov ( C N , i , C N , j ) − 1 { i = j } λ i ≤ C B N ij q i + j . (39) Proof. Separating the identical-cycle term gives
Cov ( C N , i , C N , j ) = 1 { i = j } ( μ N , i − ∑ α ∈ C N , i p α 2 ) + ∑ α ∈ C N , i , β ∈ C N , j β ≠ α p α { P ( I β = 1 ∣ I α = 1 ) − p β } . \begin{aligned}
\operatorname{Cov}(C_{N,i},C_{N,j})
={}&\mathbf{1}_{\{i=j\}}\left(\mu_{N,i}-\sum_{\alpha\in\mathcal{C}_{N,i}}p_\alpha^2\right)\\
&+\sum_{\substack{\alpha\in\mathcal{C}_{N,i},\ \beta\in\mathcal{C}_{N,j}\\\beta\ne\alpha}}p_\alpha\left\{\mathbb{P}(I_\beta=1\mid I_\alpha=1)-p_\beta\right\}.
\end{aligned} Cov ( C N , i , C N , j ) = 1 { i = j } μ N , i − α ∈ C N , i ∑ p α 2 + α ∈ C N , i , β ∈ C N , j β = α ∑ p α { P ( I β = 1 ∣ I α = 1 ) − p β } . The contribution of vertex-disjoint pairs is bounded in absolute value by 4 i j μ N , i μ N , j / N 4ij\mu_{N,i}\mu_{N,j}/N 4 ij μ N , i μ N , j / N Lemma 3.4 . For intersecting pairs, use the sum of the conditional and unconditional probabilities. Lemma 3.6 and Proposition 3.5 bound this contribution by
i j N μ N , i μ N , j + i 2 j q j N μ N , i . \frac{ij}{N}\mu_{N,i}\mu_{N,j}+\frac{i^{2}jq^{j}}{N}\mu_{N,i}. N ij μ N , i μ N , j + N i 2 j q j μ N , i . Since μ N , i ≤ λ i ≤ C B q i / i \mu_{N,i}\leq\lambda_i\leq C_Bq^i/i μ N , i ≤ λ i ≤ C B q i / i C B i j q i + j / N C_Bijq^{i+j}/N C B ij q i + j / N i = j i=j i = j Proposition 3.1 gives ∣ μ N , i − λ i ∣ ≤ C B i q i / N |\mu_{N,i}-\lambda_i|\leq C_Biq^i/N ∣ μ N , i − λ i ∣ ≤ C B i q i / N
∑ α ∈ C N , i p α 2 ≤ ( N − R ) − 3 μ N , i . \sum_{\alpha\in\mathcal{C}_{N,i}}p_\alpha^2\leq(N-R)^{-3}\mu_{N,i}. α ∈ C N , i ∑ p α 2 ≤ ( N − R ) − 3 μ N , i . When i = j i=j i = j (39) .
A second moment for the non-cycle remainder
We first record a deterministic counting fact. For a vertex set A A A d d d k k k A A A
∣ A ∣ k d q k − 1 . (40) |A|kdq^{k-1}. \tag*{(40)} ∣ A ∣ k d q k − 1 . (40) Indeed, root such a walk at a visit to A A A ∣ A ∣ d q k − 1 |A|dq^{k-1} ∣ A ∣ d q k − 1 k k k
Proposition 5.3 (Second-moment remainder bound). For k ≥ 3 k\geq3 k ≥ 3 N ≥ 4 k N\geq4k N ≥ 4 k
E [ R N , k 2 ] ≤ C B q 2 k ( k 6 N + k 8 N 2 ) . (41) \mathbb{E}[R_{N,k}^{2}]\leq C_Bq^{2k}\left(\frac{k^{6}}{N}+\frac{k^{8}}{N^{2}}\right). \tag*{(41)} E [ R N , k 2 ] ≤ C B q 2 k ( N k 6 + N 2 k 8 ) . (41) Consequently, for N ≥ 8 k 2 N\geq8k^{2} N ≥ 8 k 2
E [ R N , k 2 ] ≤ C B k 6 q 2 k N , ∥ q − k / 2 2 k ( R N , k − E [ R N , k ] ) ∥ L 2 ≤ C B ( k 5 q k N ) 1 / 2 . (42) \mathbb{E}[R_{N,k}^{2}]\leq C_B\frac{k^{6}q^{2k}}{N},\qquad
\left\|\frac{q^{-k/2}}{\sqrt{2k}}\left(R_{N,k}-\mathbb{E}[R_{N,k}]\right)\right\|_{L^{2}}
\leq C_B\left(\frac{k^{5}q^{k}}{N}\right)^{1/2}. \tag*{(42)} E [ R N , k 2 ] ≤ C B N k 6 q 2 k , 2 k q − k /2 ( R N , k − E [ R N , k ] ) L 2 ≤ C B ( N k 5 q k ) 1/2 . (42) Proof. Expand R N , k 2 R_{N,k}^{2} R N , k 2 R N , k R_{N,k} R N , k
Intersecting pairs. The first walk visits at most k k k (40) , at most k 2 d q k − 1 k^{2}dq^{k-1} k 2 d q k − 1 k k k k 2 d q k − 1 R N , k k^{2}dq^{k-1}R_{N,k} k 2 d q k − 1 R N , k Lemma 5.1 bounds its expectation by
C B k 6 q 2 k N . (43) C_B\frac{k^{6}q^{2k}}{N}. \tag*{(43)} C B N k 6 q 2 k . (43) Two intersecting remainder walks may traverse the same graph of cycle rank two, so four separate collisions need not occur.
Vertex-disjoint pairs. Index the possible first trajectories by a lifted starting vertex and a prescribed non-backtracking base walk of length k k k τ \tau τ F τ \mathcal{F}_{\tau} F τ I τ I_{\tau} I τ { I τ = 1 } \{I_{\tau}=1\} { I τ = 1 } F τ \mathcal{F}_{\tau} F τ S S S k k k k k k § 3.3 , conditional on F τ \mathcal{F}_{\tau} F τ V ( S ) V(S) V ( S ) k k k V ( S ) V(S) V ( S ) S S S S S S
Let H τ , t \mathcal{H}_{\tau,t} H τ , t F τ \mathcal{F}_{\tau} F τ t t t ξ t \xi_t ξ t ξ t = 0 \xi_t=0 ξ t = 0 2 k 2k 2 k k k k
E [ ξ t ∣ H τ , t − 1 ] ≤ p , p = k N − 2 k . \mathbb{E}[\xi_t \mid\mathcal{H}_{\tau,t-1}] \leq p,\qquad p=\frac{k}{N-2k}. E [ ξ t ∣ H τ , t − 1 ] ≤ p , p = N − 2 k k . For a < b a<b a < b ξ a \xi_a ξ a
E [ ξ a ξ b ∣ F τ ] = E [ ξ a E [ ξ b ∣ H τ , b − 1 ] ∣ F τ ] ≤ p E [ ξ a ∣ F τ ] ≤ p 2 . \mathbb{E}[\xi_a\xi_b\mid\mathcal{F}_{\tau}]
=\mathbb{E}\left[\xi_a\mathbb{E}[\xi_b\mid\mathcal{H}_{\tau,b-1}]\mid\mathcal{F}_{\tau}\right]
\leq p\mathbb{E}[\xi_a\mid\mathcal{F}_{\tau}]\leq p^2. E [ ξ a ξ b ∣ F τ ] = E [ ξ a E [ ξ b ∣ H τ , b − 1 ] ∣ F τ ] ≤ p E [ ξ a ∣ F τ ] ≤ p 2 . A disjoint remainder trajectory has ∑ t = 1 k ξ t ≥ 2 \sum_{t=1}^{k}\xi_t\geq2 ∑ t = 1 k ξ t ≥ 2 1 { ∑ t ξ t ≥ 2 } ≤ ∑ a < b ξ a ξ b \mathbf{1}_{\{\sum_t\xi_t\geq2\}}\leq\sum_{a<b}\xi_a\xi_b 1 { ∑ t ξ t ≥ 2 } ≤ ∑ a < b ξ a ξ b
n 0 N d q k − 1 ( k 2 ) ( k N − 2 k ) 2 ≤ C B k 4 q k N . n_0Ndq^{k-1}\binom{k}{2}\left(\frac{k}{N-2k}\right)^2
\leq C_B\frac{k^4q^k}{N}. n 0 N d q k − 1 ( 2 k ) ( N − 2 k k ) 2 ≤ C B N k 4 q k . This conditional bound is uniform over all transcripts with I τ = 1 I_{\tau}=1 I τ = 1 I τ I_{\tau} I τ τ \tau τ ∑ τ I τ = R N , k \sum_{\tau}I_{\tau}=R_{N,k} ∑ τ I τ = R N , k
C B k 4 q k N E [ R N , k ] ≤ C B k 8 q 2 k N 2 . C_B\frac{k^4q^k}{N}\mathbb{E}[R_{N,k}]
\leq C_B\frac{k^8q^{2k}}{N^2}. C B N k 4 q k E [ R N , k ] ≤ C B N 2 k 8 q 2 k . Together with (5.9), this proves (5.7). Under N ≥ 8 k 2 N\geq8k^2 N ≥ 8 k 2 Var ( R N , k ) ≤ E [ R N , k 2 ] \operatorname{Var}(R_{N,k})\leq\mathbb{E}[R_{N,k}^2] Var ( R N , k ) ≤ E [ R N , k 2 ]
Mixed cycle–remainder moments
The mixed terms can be estimated directly using nonnegativity and conditional cycle counts.
Proposition 5.4 (Mixed cycle–remainder bound). For integers i , ℓ ≥ 3 i,\ell\geq3 i , ℓ ≥ 3 N ≥ 2 ( i + ℓ ) N\geq2(i+\ell) N ≥ 2 ( i + ℓ )
∣ Cov ( C N , i , R N , ℓ ) ∣ ≤ C B ℓ 5 q i + ℓ N . (44) \left|\operatorname{Cov}(C_{N,i},R_{N,\ell})\right|
\leq C_B\frac{\ell^5q^{i+\ell}}{N}. \tag*{(44)} ∣ Cov ( C N , i , R N , ℓ ) ∣ ≤ C B N ℓ 5 q i + ℓ . (44) The same upper bound holds for E [ C N , i R N , ℓ ] \mathbb{E}[C_{N,i}R_{N,\ell}] E [ C N , i R N , ℓ ]
Proof. Index the possible remainder trajectories of length ℓ \ell ℓ τ \tau τ F τ \mathcal{F}_{\tau} F τ I τ I_{\tau} I τ { I τ = 1 } \{I_{\tau}=1\} { I τ = 1 } S S S ∣ V ( S ) ∣ , ∣ E ( S ) ∣ ≤ ℓ |V(S)|,|E(S)|\leq\ell ∣ V ( S ) ∣ , ∣ E ( S ) ∣ ≤ ℓ i i i V ( S ) V(S) V ( S )
Cycles meeting the support. In every d d d A A A i i i A A A ∣ A ∣ d q i − 1 / 2 |A|dq^{i-1}/2 ∣ A ∣ d q i − 1 /2 A A A i i i A A A ∣ A ∣ d q i − 1 |A|dq^{i-1} ∣ A ∣ d q i − 1 i i i
ordered pairs consisting of a remainder ℓ \ell ℓ i i i
ℓ d q i − 1 2 R N , ℓ . \frac{\ell d q^{i-1}}{2}\mathcal{R}_{N,\ell}. 2 ℓ d q i − 1 R N , ℓ . Cycles disjoint from the support. Condition on a transcript with I τ = 1 I_\tau=1 I τ = 1 S S S x 0 x_0 x 0 V ( S ) V(S) V ( S ) w w w i i i H τ , w \mathcal{H}_{\tau,w} H τ , w i − 1 i-1 i − 1 E τ , w E_{\tau,w} E τ , w x 0 , … , x i − 1 x_0,\ldots,x_{i-1} x 0 , … , x i − 1 V ( S ) V(S) V ( S ) w w w H τ , w \mathcal{H}_{\tau,w} H τ , w
On E τ , w E_{\tau,w} E τ , w i − 1 i-1 i − 1 S S S r τ , w ≤ ℓ + i − 1 r_{\tau,w}\le\ell+i-1 r τ , w ≤ ℓ + i − 1 J τ , w J_{\tau,w} J τ , w i i i S S S
E [ J τ , w ∣ H τ , w ] = 1 E τ , w N − r τ , w ≤ 1 E τ , w N − ℓ − i . \mathbb{E}[J_{\tau,w}\mid\mathcal{H}_{\tau,w}]
=\frac{\mathbf{1}_{E_{\tau,w}}}{N-r_{\tau,w}}
\le\frac{\mathbf{1}_{E_{\tau,w}}}{N-\ell-i}. E [ J τ , w ∣ H τ , w ] = N − r τ , w 1 E τ , w ≤ N − ℓ − i 1 E τ , w . We condition only on the observed prefix. Each present simple i i i S S S 2 i 2i 2 i
n 0 N d q i − 1 2 i ( N − ℓ − i ) ≤ C B q i i . \frac{n_0Ndq^{i-1}}{2i(N-\ell-i)}
\le C_B\frac{q^i}{i}. 2 i ( N − ℓ − i ) n 0 N d q i − 1 ≤ C B i q i . This bound is uniform in the first transcript, since N ≥ 2 ( i + ℓ ) N\ge2(i+\ell) N ≥ 2 ( i + ℓ ) I τ I_\tau I τ τ \tau τ C B q i E [ R N , ℓ ] / i C_Bq^i\mathbb{E}[\mathcal{R}_{N,\ell}]/i C B q i E [ R N , ℓ ] / i
Combining both contributions and using Lemma 5.1 yields
E [ C N , i R N , ℓ ] ≤ C B ( ℓ + 1 i ) q i E [ R N , ℓ ] ≤ C B ℓ 5 q i + ℓ N . \mathbb{E}[C_{N,i}\mathcal{R}_{N,\ell}]
\le C_B\left(\ell+\frac{1}{i}\right)q^i\mathbb{E}[\mathcal{R}_{N,\ell}]
\le C_B\frac{\ell^5q^{i+\ell}}{N}. E [ C N , i R N , ℓ ] ≤ C B ( ℓ + i 1 ) q i E [ R N , ℓ ] ≤ C B N ℓ 5 q i + ℓ . Also, Proposition 3.1 gives μ N , i ≤ C B q i / i \mu_{N,i}\le C_Bq^i/i μ N , i ≤ C B q i / i
μ N , i E [ R N , ℓ ] ≤ C B ℓ 4 q i + ℓ i N . \mu_{N,i}\mathbb{E}[\mathcal{R}_{N,\ell}]
\le C_B\frac{\ell^4q^{i+\ell}}{iN}. μ N , i E [ R N , ℓ ] ≤ C B i N ℓ 4 q i + ℓ . Both variables are nonnegative, so ∣ Cov ( C N , i , R N , ℓ ) ∣ ≤ E [ C N , i R N , ℓ ] + μ N , i E [ R N , ℓ ] |\operatorname{Cov}(C_{N,i},\mathcal{R}_{N,\ell})|\le\mathbb{E}[C_{N,i}\mathcal{R}_{N,\ell}]+\mu_{N,i}\mathbb{E}[\mathcal{R}_{N,\ell}] ∣ Cov ( C N , i , R N , ℓ ) ∣ ≤ E [ C N , i R N , ℓ ] + μ N , i E [ R N , ℓ ]
Pointwise polynomial-trace covariances
Define
X N , k = Y N , k 2 k , K B ( k , ℓ ) = Λ B ( k , ℓ ) 2 k ℓ . X_{N,k}=\frac{Y_{N,k}}{\sqrt{2k}},\qquad K_B(k,\ell)=\frac{\Lambda_B(k,\ell)}{2\sqrt{k\ell}}. X N , k = 2 k Y N , k , K B ( k , ℓ ) = 2 k ℓ Λ B ( k , ℓ ) . Set
Q N , k = 2 k q − k / 2 ∑ i ∣ k i ≥ 3 i ( C N , i − μ N , i ) , D N , k = X N , k − Q N , k . (45) Q_{N,k}=\sqrt{\frac{2}{k}}q^{-k/2}\sum_{\substack{i\mid k\\i\ge3}}i(C_{N,i}-\mu_{N,i}),\qquad D_{N,k}=X_{N,k}-Q_{N,k}. \tag*{(45)} Q N , k = k 2 q − k /2 i ∣ k i ≥ 3 ∑ i ( C N , i − μ N , i ) , D N , k = X N , k − Q N , k . (45) Both variables are centred. For k ≥ 3 k \ge3 k ≥ 3 N ≥ 8 k 2 N \ge8k^2 N ≥ 8 k 2
∥ D N , k ∥ L 2 ≤ C B ( k 5 q k N ) 1 / 2 . (46) \lVert D_{N,k}\rVert_{L^2} \le C_B\left(\frac{k^5q^k}{N}\right)^{1/2}. \tag*{(46)} ∥ D N , k ∥ L 2 ≤ C B ( N k 5 q k ) 1/2 . (46) Independence and Var ( Z j ) = λ j = b j / ( 2 j ) \operatorname{Var}(Z_j)=\lambda_j=b_j/(2j) Var ( Z j ) = λ j = b j / ( 2 j )
Cov ( Y k 2 k , Y ℓ 2 ℓ ) = 2 q − ( k + ℓ ) / 2 k ℓ ∑ j ∣ gcd ( k , ℓ ) j ≥ 3 j 2 λ j = K B ( k , ℓ ) . \begin{aligned}
\operatorname{Cov}\left(\frac{Y_k}{\sqrt{2k}},\frac{Y_\ell}{\sqrt{2\ell}}\right)
=\frac{2q^{-(k+\ell)/2}}{\sqrt{k\ell}}
\sum_{\substack{j\mid\gcd(k,\ell)\\j\ge3}}j^2\lambda_j \\
&=K_B(k,\ell).
\end{aligned} Cov ( 2 k Y k , 2 ℓ Y ℓ ) = k ℓ 2 q − ( k + ℓ ) /2 j ∣ g c d ( k , ℓ ) j ≥ 3 ∑ j 2 λ j = K B ( k , ℓ ) . Moreover, K B ( k , k ) ≤ C B K_B(k,k)\le C_B K B ( k , k ) ≤ C B k k k ∑ j ≤ k j b j ≤ C B k q k \sum_{j\le k}jb_j\le C_Bkq^k ∑ j ≤ k j b j ≤ C B k q k
Lemma 5.5 (Covariance of the cycle part). For 3 ≤ k , ℓ ≤ R 3\le k,\ell\le R 3 ≤ k , ℓ ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
∣ Cov ( Q N , k , Q N , ℓ ) − K B ( k , ℓ ) ∣ ≤ C B k 3 / 2 ℓ 3 / 2 q ( k + ℓ ) / 2 N . \left|\operatorname{Cov}(Q_{N,k},Q_{N,\ell})-K_B(k,\ell)\right|
\le C_B\frac{k^{3/2}\ell^{3/2}q^{(k+\ell)/2}}{N}. ∣ Cov ( Q N , k , Q N , ℓ ) − K B ( k , ℓ ) ∣ ≤ C B N k 3/2 ℓ 3/2 q ( k + ℓ ) /2 . Proof. Apply Proposition 5.2 term by term in (5.11). The independent-Poisson part is exactly K B ( k , ℓ ) K_B(k,\ell) K B ( k , ℓ )
C B q − ( k + ℓ ) / 2 N k ℓ ( ∑ i ∣ k i 2 q i ) ( ∑ j ∣ ℓ j 2 q j ) . \frac{C_Bq^{-(k+\ell)/2}}{N\sqrt{k\ell}}
\left(\sum_{i\mid k}i^2q^i\right)
\left(\sum_{j\mid\ell}j^2q^j\right). N k ℓ C B q − ( k + ℓ ) /2 i ∣ k ∑ i 2 q i j ∣ ℓ ∑ j 2 q j . The geometric bound ∑ i ≤ k i 2 q i ≤ C q k 2 q k \sum_{i\le k}i^2q^i\le C_qk^2q^k ∑ i ≤ k i 2 q i ≤ C q k 2 q k ℓ \ell ℓ
Proposition 5.6 (Pointwise covariance estimate). For 3 ≤ k , ℓ ≤ R 3\le k,\ell\le R 3 ≤ k , ℓ ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
∣ Cov ( Y N , k , Y N , ℓ ) − Λ B ( k , ℓ ) ∣ ≤ C B q ( k + ℓ ) / 2 N ( k 2 ℓ 2 + k ℓ 5 + k 5 ℓ + k 3 ℓ 3 ) . (47) \left|\operatorname{Cov}(Y_{N,k},Y_{N,\ell})-\Lambda_B(k,\ell)\right|
\le\frac{C_Bq^{(k+\ell)/2}}{N}
\left(k^2\ell^2+k\ell^5+k^5\ell+k^3\ell^3\right). \tag*{(47)} ∣ Cov ( Y N , k , Y N , ℓ ) − Λ B ( k , ℓ ) ∣ ≤ N C B q ( k + ℓ ) /2 ( k 2 ℓ 2 + k ℓ 5 + k 5 ℓ + k 3 ℓ 3 ) . (47) Primitivity is not required.
Proof. In the notation of (5.11), decompose the normalised covariance as
Cov ( X N , k , X N , ℓ ) − K B ( k , ℓ ) = Cov ( Q N , k , Q N , ℓ ) − K B ( k , ℓ ) + Cov ( Q N , k , D N , ℓ ) + Cov ( D N , k , Q N , ℓ ) + Cov ( D N , k , D N , ℓ ) . \begin{aligned}
\operatorname{Cov}(X_{N,k},X_{N,\ell})-K_B(k,\ell)
={}&\operatorname{Cov}(Q_{N,k},Q_{N,\ell})-K_B(k,\ell)\\
&+\operatorname{Cov}(Q_{N,k},D_{N,\ell})
+\operatorname{Cov}(D_{N,k},Q_{N,\ell})\\
&+\operatorname{Cov}(D_{N,k},D_{N,\ell}).
\end{aligned} Cov ( X N , k , X N , ℓ ) − K B ( k , ℓ ) = Cov ( Q N , k , Q N , ℓ ) − K B ( k , ℓ ) + Cov ( Q N , k , D N , ℓ ) + Cov ( D N , k , Q N , ℓ ) + Cov ( D N , k , D N , ℓ ) . Lemma 5.5 controls the first term by C B k 3 / 2 ℓ 3 / 2 q ( k + ℓ ) / 2 / N C_Bk^{3/2}\ell^{3/2}q^{(k+\ell)/2}/N C B k 3/2 ℓ 3/2 q ( k + ℓ ) /2 / N D N , ℓ = q − ℓ / 2 ( R N , ℓ − E [ R N , ℓ ] ) / 2 ℓ D_{N,\ell}=q^{-\ell/2}(R_{N,\ell}-\mathbb{E}[R_{N,\ell}])/\sqrt{2\ell} D N , ℓ = q − ℓ /2 ( R N , ℓ − E [ R N , ℓ ]) / 2 ℓ i ∣ k i\mid k i ∣ k N ≥ 8 R 2 ≥ 2 ( i + ℓ ) N\ge8R^2\ge2(i+\ell) N ≥ 8 R 2 ≥ 2 ( i + ℓ )
∣ Cov ( Q N , k , D N , ℓ ) ∣ ≤ q − ( k + ℓ ) / 2 k ℓ ∑ i ∣ k i ≥ 3 i ∣ Cov ( C N , i , R N , ℓ ) ∣ \left|\operatorname{Cov}(Q_{N,k},D_{N,\ell})\right|
\le\frac{q^{-(k+\ell)/2}}{\sqrt{k\ell}}
\sum_{\substack{i\mid k\\i\ge3}}i\left|\operatorname{Cov}(C_{N,i},R_{N,\ell})\right| ∣ Cov ( Q N , k , D N , ℓ ) ∣ ≤ k ℓ q − ( k + ℓ ) /2 i ∣ k i ≥ 3 ∑ i ∣ Cov ( C N , i , R N , ℓ ) ∣ ≤ C B ℓ 5 q ( ℓ − k ) / 2 N k ℓ ∑ i ∣ k i q i \le\frac{C_B\ell^5q^{(\ell-k)/2}}{N\sqrt{k\ell}}
\sum_{i\mid k}iq^i ≤ N k ℓ C B ℓ 5 q ( ℓ − k ) /2 i ∣ k ∑ i q i ≤ C B k 1 / 2 ℓ 9 / 2 q ( k + ℓ ) / 2 N . \le C_B\frac{k^{1/2}\ell^{9/2}q^{(k+\ell)/2}}{N}. ≤ C B N k 1/2 ℓ 9/2 q ( k + ℓ ) /2 . Here ∑ i ≤ k i q i ≤ C q k q k \sum_{i\le k}iq^i\le C_qkq^k ∑ i ≤ k i q i ≤ C q k q k k k k ℓ \ell ℓ
∣ Cov ( D N , k , D N , ℓ ) ∣ ≤ C B k 5 / 2 ℓ 5 / 2 q ( k + ℓ ) / 2 N . \left|\operatorname{Cov}(D_{N,k},D_{N,\ell})\right|
\le C_B\frac{k^{5/2}\ell^{5/2}q^{(k+\ell)/2}}{N}. ∣ Cov ( D N , k , D N , ℓ ) ∣ ≤ C B N k 5/2 ℓ 5/2 q ( k + ℓ ) /2 . Multiply each of the four normalised bounds by 2 k 2 ℓ \sqrt{2k}\sqrt{2\ell} 2 k 2 ℓ
The same normalised bounds also imply
sup 3 ≤ k , ℓ ≤ R ∣ Cov ( X N , k , X N , ℓ ) − K B ( k , ℓ ) ∣ ≤ C B R 5 q R / N . \sup_{3\le k,\ell\le R}\left|\operatorname{Cov}(X_{N,k},X_{N,\ell})-K_B(k,\ell)\right|\le C_B R^5q^R/N. 3 ≤ k , ℓ ≤ R sup ∣ Cov ( X N , k , X N , ℓ ) − K B ( k , ℓ ) ∣ ≤ C B R 5 q R / N . In particular, for every fixed L L L
Cov ( Y N , k , Y N , ℓ ) = Λ B ( k , ℓ ) + O B , L ( N − 1 ) , 3 ≤ k , ℓ ≤ L . (48) \operatorname{Cov}(Y_{N,k},Y_{N,\ell})=\Lambda_B(k,\ell)+O_{B,L}(N^{-1}),\qquad3\le k,\ell\le L. \tag*{(48)} Cov ( Y N , k , Y N , ℓ ) = Λ B ( k , ℓ ) + O B , L ( N − 1 ) , 3 ≤ k , ℓ ≤ L . (48) The growing-time transfer uses the pointwise estimate (47) , rather than only its uniform cutoff bound. The covariance scale alone can also be recovered from the core-graph method discussed in § 1 ; the proof above uses only the explicit exploration estimates in this paper.
An oscillatory Chebyshev energy limit
Set
ϕ ( x ) = arccos ( r d cos x ) , f s ( x ) = T s ( r d cos x ) = cos ( s ϕ ( x ) ) , x ∈ R / ( 2 π Z ) . \phi(x)=\arccos(r_d\cos x),\qquad f_s(x)=T_s(r_d\cos x)=\cos(s\phi(x)),\qquad x\in\mathbb{R}/(2\pi\mathbb{Z}). ϕ ( x ) = arccos ( r d cos x ) , f s ( x ) = T s ( r d cos x ) = cos ( s ϕ ( x )) , x ∈ R / ( 2 π Z ) . The phase is smooth and periodic because 0 < r d < 1 0<r_d<1 0 < r d < 1 ( 0 , π ) (0,\pi) ( 0 , π )
ϕ ( 0 ) = α d , ϕ ( π ) = π − α d , ϕ ′ ( x ) = r d sin x 1 − r d 2 cos 2 x . \phi(0)=\alpha_d,\qquad\phi(\pi)=\pi-\alpha_d,\qquad\phi'(x)=\frac{r_d\sin x}{\sqrt{1-r_d^2\cos^2x}}. ϕ ( 0 ) = α d , ϕ ( π ) = π − α d , ϕ ′ ( x ) = 1 − r d 2 cos 2 x r d sin x . The coefficients in (12) are the cosine coefficients of f s f_s f s 2 π 2\pi 2 π L 1 L^1 L 1 ∥ g ∥ L 1 = ∫ 0 2 π ∣ g ( x ) ∣ d x \|g\|_{L^1}=\int_0^{2\pi}|g(x)|\,\mathrm{d}x ∥ g ∥ L 1 = ∫ 0 2 π ∣ g ( x ) ∣ d x ∥ g ∥ ∞ = sup x ∈ [ 0 , 2 π ] ∣ g ( x ) ∣ \|g\|_\infty=\sup_{x\in[0,2\pi]}|g(x)| ∥ g ∥ ∞ = sup x ∈ [ 0 , 2 π ] ∣ g ( x ) ∣ 2 π 2\pi 2 π
Lemma 6.1 (Uniform coefficient bounds). For every $s\ge 0,
a s , 0 2 + 1 2 ∑ k = 1 s a s , k 2 = 1 π ∫ 0 π ∣ f s ( x ) ∣ 2 d x ≤ 1. (49) a_{s,0}^2+\frac{1}{2}\sum_{k=1}^{s}a_{s,k}^2=\frac{1}{\pi}\int_0^\pi|f_s(x)|^2\,\mathrm{d}x\le1. \tag*{(49)} a s , 0 2 + 2 1 k = 1 ∑ s a s , k 2 = π 1 ∫ 0 π ∣ f s ( x ) ∣ 2 d x ≤ 1. (49) In particular, ∣ a s , k ∣ ≤ 2 |a_{s,k}|\le\sqrt{2} ∣ a s , k ∣ ≤ 2 k ≥ 1 k\ge1 k ≥ 1 For each fixed p ≥ 0 p\ge0 p ≥ 0 and s ≤ R s\le R s ≤ R
∑ k = 1 s ∣ a s , k ∣ k p q k / 2 ≤ C q R p q R / 2 , ∑ k = 1 s k a s , k 2 ≤ 2 s . (50) \sum_{k=1}^{s}|a_{s,k}|k^pq^{k/2}\le C_qR^pq^{R/2},\qquad\sum_{k=1}^{s}ka_{s,k}^2\le2s. \tag*{(50)} k = 1 ∑ s ∣ a s , k ∣ k p q k /2 ≤ C q R p q R /2 , k = 1 ∑ s k a s , k 2 ≤ 2 s . (50) Also a s , k = 0 a_{s,k}=0 a s , k = 0 s s s k k k
Proof. Parseval and ∣ cos ( s ϕ ) ∣ ≤ 1 |\cos(s\phi)|\le1 ∣ cos ( s ϕ ) ∣ ≤ 1 (49) . Cauchy–Schwarz and ∑ k = 1 R k 2 p q k ≤ R 2 p q R / ( 1 − q − 1 ) \sum_{k=1}^{R}k^{2p}q^k\le R^{2p}q^R/(1-q^{-1}) ∑ k = 1 R k 2 p q k ≤ R 2 p q R / ( 1 − q − 1 ) (50) ; k ≤ s k\le s k ≤ s T s ( − r d x ) = ( − 1 ) s T s ( r d x ) T_s(-r_dx)=(-1)^sT_s(r_dx) T s ( − r d x ) = ( − 1 ) s T s ( r d x )
The two stationary points 0 , π 0,\pi 0 , π
ϕ ′ ′ ( x ) = r d ( 1 − r d 2 ) cos x ( 1 − r d 2 cos 2 x ) 3 / 2 , ϕ ′ ′ ( 0 ) = r d 1 − r d 2 , ϕ ′ ′ ( π ) = − r d 1 − r d 2 . \phi''(x)=\frac{r_d(1-r_d^2)\cos x}{(1-r_d^2\cos^2x)^{3/2}},\qquad\phi''(0)=\frac{r_d}{\sqrt{1-r_d^2}},\qquad\phi''(\pi)=-\frac{r_d}{\sqrt{1-r_d^2}}. ϕ ′′ ( x ) = ( 1 − r d 2 cos 2 x ) 3/2 r d ( 1 − r d 2 ) cos x , ϕ ′′ ( 0 ) = 1 − r d 2 r d , ϕ ′′ ( π ) = − 1 − r d 2 r d . Lemma 6.2 (Uniform oscillatory estimate). For a continuously differentiable periodic complex-valued amplitude A A A and ω ≥ 1 \omega\ge1 ω ≥ 1
∣ ∫ 0 2 π e i ω ϕ ( x ) A ( x ) d x ∣ ≤ C r d ω − 1 / 2 ( ∥ A ∥ ∞ + ∥ A ′ ∥ L 1 ) . \left|\int_0^{2\pi}e^{i\omega\phi(x)}A(x)\,\mathrm{d}x\right|\le C r_d\omega^{-1/2}\left(\|A\|_\infty+\|A'\|_{L^1}\right). ∫ 0 2 π e iω ϕ ( x ) A ( x ) d x ≤ C r d ω − 1/2 ( ∥ A ∥ ∞ + ∥ A ′ ∥ L 1 ) . Proof. For large ω \omega ω ρ = ω − 1 / 2 \rho=\omega^{-1/2} ρ = ω − 1/2 0 0 0 π \pi π C ρ ∥ A ∥ ∞ C\rho\|A\|_{\infty} C ρ ∥ A ∥ ∞ ∣ ϕ ′ ∣ ≥ c r d ρ |\phi'|\ge c_{r_d}\rho ∣ ϕ ′ ∣ ≥ c r d ρ ∣ ϕ ′ ∣ ≥ r d ∣ sin x ∣ |\phi'|\ge r_d|\sin x| ∣ ϕ ′ ∣ ≥ r d ∣ sin x ∣ ϕ ′ ′ \phi'' ϕ ′′ ∣ sin x ∣ − 2 |\sin x|^{-2} ∣ sin x ∣ − 2 4 cot ρ ≤ C / ρ 4\cot\rho\le C/\rho 4 cot ρ ≤ C / ρ ρ \rho ρ
∫ dist ( x , { 0 , π } ) ≥ ρ ∣ ϕ ′ ′ ( x ) ∣ ∣ ϕ ′ ( x ) ∣ 2 d x ≤ C r d / ρ . \int_{\operatorname{dist}(x,\{0,\pi\})\ge\rho}\frac{|\phi''(x)|}{|\phi'(x)|^2}\,dx\le C_{r_d}/\rho. ∫ dist ( x , { 0 , π }) ≥ ρ ∣ ϕ ′ ( x ) ∣ 2 ∣ ϕ ′′ ( x ) ∣ d x ≤ C r d / ρ . Integrate by parts on each complementary interval, using e i ω ϕ = ( i ω ϕ ′ ) − 1 ( e i ω ϕ ) ′ e^{i\omega\phi}=(i\omega\phi')^{-1}(e^{i\omega\phi})' e iω ϕ = ( iω ϕ ′ ) − 1 ( e iω ϕ ) ′ A ϕ ′ ′ / ( ϕ ′ ) 2 A\phi''/(\phi')^2 A ϕ ′′ / ( ϕ ′ ) 2 C r d ∥ A ∥ ∞ / ( ω ρ ) C_{r_d}\|A\|_{\infty}/(\omega\rho) C r d ∥ A ∥ ∞ / ( ω ρ ) A ′ / ϕ ′ A'/\phi' A ′ / ϕ ′ C r d ∥ A ′ ∥ 1 / ( ω ρ ) C_{r_d}\|A'\|_1/(\omega\rho) C r d ∥ A ′ ∥ 1 / ( ω ρ ) ω \omega ω C r d C_{r_d} C r d
Bilinear energy and observation-time normalisation
The leading covariance term scales with the geometric mean of the two observation times. We begin with the Fourier identity used in the proof.
Write f s ( x ) = cos ( s ϕ ( x ) ) f_s(x)=\cos(s\phi(x)) f s ( x ) = cos ( s ϕ ( x )) Δ u f ( x ) = f ( x + u ) − f ( x ) \Delta_u f(x)=f(x+u)-f(x) Δ u f ( x ) = f ( x + u ) − f ( x ) f , g f,g f , g
f ^ ( n ) = 1 2 π ∫ 0 2 π f ( x ) e − i n x d x . \widehat f(n)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx. f ( n ) = 2 π 1 ∫ 0 2 π f ( x ) e − in x d x . The homogeneous H 1 / 2 H^{1/2} H 1/2
E ( f , g ) = ∑ n ∈ Z ∣ n ∣ f ^ ( n ) g ^ ( n ) ‾ = 1 16 π 2 ∫ 0 2 π ∫ − π π Δ u f ( x ) Δ u g ( x ) sin 2 ( u / 2 ) d u d x . \mathcal{E}(f,g)=\sum_{n\in\mathbb{Z}}|n|\widehat f(n)\overline{\widehat g(n)}
=\frac{1}{16\pi^2}\int_0^{2\pi}\int_{-\pi}^{\pi}
\frac{\Delta_u f(x)\Delta_u g(x)}{\sin^2(u/2)}\,du\,dx. E ( f , g ) = n ∈ Z ∑ ∣ n ∣ f ( n ) g ( n ) = 16 π 2 1 ∫ 0 2 π ∫ − π π sin 2 ( u /2 ) Δ u f ( x ) Δ u g ( x ) d u d x . This identity follows from Fourier orthogonality: it can first be verified for trigonometric polynomials and then extended by approximation. Since each f s f_s f s
E ( f s , f s ′ ) = 1 2 ∑ k ≥ 1 k a s , k a s ′ , k . (51) \mathcal{E}(f_s,f_{s'})=\frac{1}{2}\sum_{k\ge1}k a_{s,k}a_{s',k}. \tag*{(51)} E ( f s , f s ′ ) = 2 1 k ≥ 1 ∑ k a s , k a s ′ , k . (51) Lemma 6.3 (Uniform absolute-derivative remainder). Let φ ∈ C ∞ ( T ; R ) \varphi\in C^\infty(\mathbb{T};\mathbb{R}) φ ∈ C ∞ ( T ; R ) have finitely many critical points, all nondegenerate. With
D = − i ∂ x , ∣ D ∣ g = ∑ k ∈ Z ∣ k ∣ g ^ ( k ) e i k x , D=-i\partial_x,\qquad|D|g=\sum_{k\in\mathbb{Z}}|k|\widehat g(k)e^{ikx}, D = − i ∂ x , ∣ D ∣ g = k ∈ Z ∑ ∣ k ∣ g ( k ) e ik x , there is C φ < ∞ C_\varphi<\infty C φ < ∞ such that, for every integer n ≥ 1 n\ge1 n ≥ 1
∥ ∣ D ∣ e i n φ − n ∣ φ ′ ∣ e i n φ ∥ L 1 ( T ) ≤ C φ . (52) \left\||D|e^{in\varphi}-n|\varphi'|e^{in\varphi}\right\|_{L^1(\mathbb{T})}\le C_\varphi. \tag*{(52)} ∣ D ∣ e in φ − n ∣ φ ′ ∣ e in φ L 1 ( T ) ≤ C φ . (52) The same estimate holds after taking real parts.
Proof. For smooth periodic g g g
∣ D ∣ g ( x ) = 1 4 π p.v. ∫ − π π g ( x ) − g ( x + v ) sin 2 ( v / 2 ) d v . (53) |D|g(x)=\frac{1}{4\pi}\operatorname{p.v.}\int_{-\pi}^{\pi}
\frac{g(x)-g(x+v)}{\sin^2(v/2)}\,dv. \tag*{(53)} ∣ D ∣ g ( x ) = 4 π 1 p.v. ∫ − π π sin 2 ( v /2 ) g ( x ) − g ( x + v ) d v . (53) ∣ D ∣ g ( x ) = 1 8 π ∫ − π π 2 g ( x ) − g ( x + v ) − g ( x − v ) sin 2 ( v / 2 ) d v . (54) |D|g(x)=\frac{1}{8\pi}\int_{-\pi}^{\pi}
\frac{2g(x)-g(x+v)-g(x-v)}{\sin^2(v/2)}\,dv. \tag*{(54)} ∣ D ∣ g ( x ) = 8 π 1 ∫ − π π sin 2 ( v /2 ) 2 g ( x ) − g ( x + v ) − g ( x − v ) d v . (54) The second integral is absolutely convergent. The identities follow first on Fourier modes from
∫ − π π 1 − cos ( k v ) sin 2 ( v / 2 ) d v = 4 π ∣ k ∣ \int_{-\pi}^{\pi}\frac{1-\cos(kv)}{\sin^2(v/2)}\,dv=4\pi|k| ∫ − π π sin 2 ( v /2 ) 1 − cos ( k v ) d v = 4 π ∣ k ∣ and then by smooth approximation. The remainder is estimated by separating neighbourhoods of the critical points from their complement. Let C = { φ ′ = 0 } C=\{\varphi'=0\} C = { φ ′ = 0 } δ ( x ) = min { δ 0 , dist T ( x , C ) } \delta(x)=\min\{\delta_0,\operatorname{dist}_{\mathbb{T}}(x,C)\} δ ( x ) = min { δ 0 , dist T ( x , C )} δ 0 > 0 \delta_0>0 δ 0 > 0 ∣ φ ′ ( x ) ∣ ≍ δ ( x ) |\varphi'(x)|\asymp\delta(x) ∣ φ ′ ( x ) ∣ ≍ δ ( x ) F n = e i n φ F_n=e^{in\varphi} F n = e in φ R n = ∣ D ∣ F n − n ∣ φ ′ ∣ F n R_n=|D|F_n-n|\varphi'|F_n R n = ∣ D ∣ F n − n ∣ φ ′ ∣ F n δ ( x ) ≤ ℓ : = n − 1 / 2 \delta(x)\le\ell:=n^{-1/2} δ ( x ) ≤ ℓ := n − 1/2 ∣ F n ′ ′ ∣ ≤ C φ n |F_n''|\le C_\varphi n ∣ F n ′′ ∣ ≤ C φ n 2 ℓ 2\ell 2 ℓ (54) for ∣ v ∣ ≤ ℓ |v|\le\ell ∣ v ∣ ≤ ℓ
∣ R n ( x ) ∣ ≤ C φ n . (55) |R_n(x)|\le C_\varphi\sqrt{n}. \tag*{(55)} ∣ R n ( x ) ∣ ≤ C φ n . (55) Suppose now that δ : = δ ( x ) > ℓ \delta:=\delta(x)>\ell δ := δ ( x ) > ℓ σ = sgn φ ′ ( x ) \sigma=\operatorname{sgn}\varphi'(x) σ = sgn φ ′ ( x ) χ ∈ C ∞ ( ( − 1 / 4 , 1 / 4 ) ) \chi\in C^\infty((-1/4,1/4)) χ ∈ C ∞ (( − 1/4 , 1/4 )) 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 χ = 1 \chi=1 χ = 1 [ − 1 / 8 , 1 / 8 ] [-1/8,1/8] [ − 1/8 , 1/8 ] x x x δ \delta δ g ( y ) = χ ( ( y − x ) / δ ) F n ( y ) g(y)=\chi((y-x)/\delta)F_n(y) g ( y ) = χ (( y − x ) / δ ) F n ( y ) b = F n − g b=F_n-g b = F n − g y y y g = F n g=F_n g = F n x x x
R n ( x ) = ( ∣ D ∣ − σ D ) g ( x ) + ∣ D ∣ b ( x ) . R_n(x)=(|D|-\sigma D)g(x)+|D|b(x). R n ( x ) = ( ∣ D ∣ − σ D ) g ( x ) + ∣ D ∣ b ( x ) . The first multiplier vanishes on Fourier modes with σ k ≥ 0 \sigma k\ge0 σ k ≥ 0 σ k < 0 \sigma k<0 σ k < 0 y = x + δ v y=x+\delta v y = x + δ v
Q = n δ 2 + ∣ k ∣ δ , ψ ( v ) = n { φ ( x + δ v ) − φ ( x ) } − k δ v Q . Q=n\delta^2+|k|\delta,\qquad\psi(v)=\frac{n\{\varphi(x+\delta v)-\varphi(x)\}-k\delta v}{Q}. Q = n δ 2 + ∣ k ∣ δ , ψ ( v ) = Q n { φ ( x + δ v ) − φ ( x )} − k δ v . On the fixed support of χ \chi χ ∣ ψ ′ ∣ ≥ c φ |\psi'|\ge c_\varphi ∣ ψ ′ ∣ ≥ c φ ∥ ψ ( j ) ∥ ∞ ≤ C φ , j \|\psi^{(j)}\|_\infty\le C_{\varphi,j} ∥ ψ ( j ) ∥ ∞ ≤ C φ , j 1 ≤ j ≤ 4 1\le j\le4 1 ≤ j ≤ 4 j ≥ 2 j\ge2 j ≥ 2 n δ j ∥ φ ( j ) ∥ ∞ / Q ≤ C φ δ j − 2 n\delta^j\|\varphi^{(j)}\|_\infty/Q\le C_\varphi\delta^{j-2} n δ j ∥ φ ( j ) ∥ ∞ / Q ≤ C φ δ j − 2 L ψ a = − ( a / ψ ′ ) L_\psi a=-(a/\psi') L ψ a = − ( a / ψ ′ )
∫ e i Q ψ ( v ) χ ( v ) d v = ( i Q ) − 3 ∫ e i Q ψ ( v ) L ψ 3 χ ( v ) d v , ∥ L ψ 3 χ ∥ 1 ≤ C φ . \int e^{iQ\psi(v)}\chi(v)\,dv=(iQ)^{-3}\int e^{iQ\psi(v)}L_\psi^3\chi(v)\,dv,\qquad\|L_\psi^3\chi\|_1\le C_\varphi. ∫ e i Q ψ ( v ) χ ( v ) d v = ( i Q ) − 3 ∫ e i Q ψ ( v ) L ψ 3 χ ( v ) d v , ∥ L ψ 3 χ ∥ 1 ≤ C φ . Consequently
∣ g ^ ( k ) ∣ ≤ C φ δ ( n δ 2 + ∣ k ∣ δ ) − 3 , σ k < 0. |\widehat{g}(k)|\le C_\varphi\delta(n\delta^2+|k|\delta)^{-3},\qquad\sigma k<0. ∣ g ( k ) ∣ ≤ C φ δ ( n δ 2 + ∣ k ∣ δ ) − 3 , σ k < 0. With A = n δ 2 ≥ 1 A=n\delta^2\ge1 A = n δ 2 ≥ 1 K = ⌊ A / δ ⌋ K=\lfloor A/\delta\rfloor K = ⌊ A / δ ⌋
δ ∑ k ≥ 1 k ( A + k δ ) 3 ≤ C δ A . \delta\sum_{k\ge1}\frac{k}{(A+k\delta)^3}\le\frac{C}{\delta A}. δ k ≥ 1 ∑ ( A + k δ ) 3 k ≤ δ A C . The contribution from k ≤ K k\le K k ≤ K δ K ( K + 1 ) / ( 2 A 3 ) ≤ C / ( δ A ) \delta K(K+1)/(2A^3)\le C/(\delta A) δ K ( K + 1 ) / ( 2 A 3 ) ≤ C / ( δ A ) k > K k>K k > K δ − 2 ∑ k > K k − 2 ≤ C / ( δ A ) \delta^{-2}\sum_{k>K}k^{-2}\le C/(\delta A) δ − 2 ∑ k > K k − 2 ≤ C / ( δ A )
∣ ( ∣ D ∣ − σ D ) g ( x ) ∣ ≤ C φ ( n δ 3 ) − 1 . (56) \left|(|D|-\sigma D)g(x)\right|\le C_\varphi(n\delta^3)^{-1}. \tag*{(56)} ∣ ( ∣ D ∣ − σ D ) g ( x ) ∣ ≤ C φ ( n δ 3 ) − 1 . (56) For the nonlocal term, b b b x x x (53) gives an ordinary integral,
∣ D ∣ b ( x ) = ∫ T e i n φ ( y ) A x , δ ( y ) d y , A x , δ ( y ) = − 1 − χ x , δ ( y ) 4 π sin 2 ( ( y − x ) / 2 ) . |D|b(x)=\int_{\mathbb{T}}e^{in\varphi(y)}A_{x,\delta}(y)\,dy,\qquad A_{x,\delta}(y)=-\frac{1-\chi_{x,\delta}(y)}{4\pi\sin^2((y-x)/2)}. ∣ D ∣ b ( x ) = ∫ T e in φ ( y ) A x , δ ( y ) d y , A x , δ ( y ) = − 4 π sin 2 (( y − x ) /2 ) 1 − χ x , δ ( y ) . Here χ x , δ \chi_{x,\delta} χ x , δ g g g χ ( ( y − x ) / δ ) \chi((y-x)/\delta) χ (( y − x ) / δ ) x x x y = x y=x y = x
∥ A x , δ ∥ ∞ + ∥ ∂ y A x , δ ∥ L 1 ≤ C δ − 2 . \|A_{x,\delta}\|_\infty+\|\partial_yA_{x,\delta}\|_{L^1}\le C\delta^{-2}. ∥ A x , δ ∥ ∞ + ∥ ∂ y A x , δ ∥ L 1 ≤ C δ − 2 . Here the kernel and its derivative are O ( dist ( x , y ) − 2 ) O(\operatorname{dist}(x,y)^{-2}) O ( dist ( x , y ) − 2 ) O ( dist ( x , y ) − 3 ) O(\operatorname{dist}(x,y)^{-3}) O ( dist ( x , y ) − 3 ) O ( δ − 1 ) O(\delta^{-1}) O ( δ − 1 ) O ( δ ) O(\delta) O ( δ ) n − 1 / 2 n^{-1/2} n − 1/2 φ \varphi φ
∣ ∫ T e i n φ ( y ) A ( y ) d y ∣ ≤ C φ n − 1 / 2 ( ∥ A ∥ ∞ + ∥ A ′ ∥ 1 ) . \left|\int_{\mathbb{T}}e^{in\varphi(y)}A(y)\,dy\right|\le C_\varphi n^{-1/2}\bigl(\|A\|_\infty+\|A'\|_1\bigr). ∫ T e in φ ( y ) A ( y ) d y ≤ C φ n − 1/2 ( ∥ A ∥ ∞ + ∥ A ′ ∥ 1 ) . Hence
∣ ∣ D ∣ b ( x ) ∣ ≤ C φ n − 1 / 2 δ − 2 . (57) \left||D|b(x)\right|\le C_\varphi n^{-1/2}\delta^{-2}. \tag*{(57)} ∣ ∣ D ∣ b ( x ) ∣ ≤ C φ n − 1/2 δ − 2 . (57) Integrating (55) over the O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) (56) –(57) over their complement gives
1 n ∫ n − 1 / 2 δ 0 v − 3 d v = O ( 1 ) , 1 n ∫ n − 1 / 2 δ 0 v − 2 d v = O ( 1 ) . \frac{1}{n}\int_{n^{-1/2}}^{\delta_0}v^{-3}\,\mathrm{d}v=O(1),\qquad\frac{1}{\sqrt{n}}\int_{n^{-1/2}}^{\delta_0}v^{-2}\,\mathrm{d}v=O(1). n 1 ∫ n − 1/2 δ 0 v − 3 d v = O ( 1 ) , n 1 ∫ n − 1/2 δ 0 v − 2 d v = O ( 1 ) . This proves (52) . Taking real parts is legitimate because ∣ D ∣ |D| ∣ D ∣
Theorem 6.4 (Uniform bilinear energy remainder). There is a constant C d < ∞ C_d<\infty C d < ∞ s , u s,u s , u
∣ 2 ∑ k ≥ 1 k a s , k a u , k − s u Γ d ( s − u ) ∣ ≤ C d . (58) \left|2\sum_{k\geq1}k a_{s,k}a_{u,k}-\sqrt{su}\,\Gamma_d(s-u)\right|\leq C_d. \tag*{(58)} 2 k ≥ 1 ∑ k a s , k a u , k − s u Γ d ( s − u ) ≤ C d . (58) Consequently, for every t ≥ 1 t\geq1 t ≥ 1
sup t ≤ s , u ≤ 2 t ∣ 2 s u ∑ k ≥ 1 k a s , k a u , k − Γ d ( s − u ) ∣ ≤ C d t . (59) \sup_{t\leq s,u\leq2t}\left|\frac{2}{\sqrt{su}}\sum_{k\geq1}k a_{s,k}a_{u,k}-\Gamma_d(s-u)\right|\leq\frac{C_d}{t}. \tag*{(59)} t ≤ s , u ≤ 2 t sup s u 2 k ≥ 1 ∑ k a s , k a u , k − Γ d ( s − u ) ≤ t C d . (59) All time indices are integers. The estimate holds for all positive observation times: the proof applies the multiplier estimate only to the u u u ∥ f s ∥ ∞ ≤ 1 \lVert f_s\rVert_\infty\leq1 ∥ f s ∥ ∞ ≤ 1 s s s u u u
Proof. For ϕ ( x ) = arccos ( r d cos x ) \phi(x)=\arccos(r_d\cos x) ϕ ( x ) = arccos ( r d cos x ) f n ( x ) = cos ( n ϕ ( x ) ) f_n(x)=\cos(n\phi(x)) f n ( x ) = cos ( n ϕ ( x ))
ϕ ′ ( x ) = r d sin x 1 − r d 2 cos 2 x . \phi'(x)=\frac{r_d\sin x}{\sqrt{1-r_d^2\cos^2x}}. ϕ ′ ( x ) = 1 − r d 2 cos 2 x r d sin x . Its only critical points are 0 0 0 π \pi π Lemma 6.3 therefore gives
∣ D ∣ f u = u ∣ ϕ ′ ∣ f u + r u , ∥ r u ∥ 1 ≤ C d . |D|f_u=u|\phi'|f_u+r_u,\qquad\lVert r_u\rVert_1\leq C_d. ∣ D ∣ f u = u ∣ ϕ ′ ∣ f u + r u , ∥ r u ∥ 1 ≤ C d . By (51) and Fourier orthogonality,
2 ∑ k ≥ 1 k a s , k a u , k = 2 π ∫ 0 2 π f s ( x ) ∣ D ∣ f u ( x ) d x . 2\sum_{k\geq1}k a_{s,k}a_{u,k}=\frac{2}{\pi}\int_0^{2\pi}f_s(x)|D|f_u(x)\,\mathrm{d}x. 2 k ≥ 1 ∑ k a s , k a u , k = π 2 ∫ 0 2 π f s ( x ) ∣ D ∣ f u ( x ) d x . Since ∥ f s ∥ ∞ ≤ 1 \lVert f_s\rVert_\infty\leq1 ∥ f s ∥ ∞ ≤ 1 θ = ϕ ( x ) \theta=\phi(x) θ = ϕ ( x ) s , u ≥ 1 s,u\geq1 s , u ≥ 1
2 ∑ k ≥ 1 k a s , k a u , k = u { Γ d ( s − u ) + Γ d ( s + u ) } + O d ( 1 ) . 2\sum_{k\geq1}k a_{s,k}a_{u,k}=u\{\Gamma_d(s-u)+\Gamma_d(s+u)\}+O_d(1). 2 k ≥ 1 ∑ k a s , k a u , k = u { Γ d ( s − u ) + Γ d ( s + u )} + O d ( 1 ) . For h ≠ 0 h\neq0 h = 0 (10) gives ∣ h Γ d ( h ) ∣ ≤ 4 / π |h\Gamma_d(h)|\leq4/\pi ∣ h Γ d ( h ) ∣ ≤ 4/ π ∣ u Γ d ( s + u ) ∣ ≤ 4 / π |u\Gamma_d(s+u)|\leq4/\pi ∣ u Γ d ( s + u ) ∣ ≤ 4/ π s ≠ u s\neq u s = u
∣ u − s u ∣ = u ∣ s − u ∣ s + u ≤ ∣ s − u ∣ , |u-\sqrt{su}|=\frac{\sqrt{u}|s-u|}{\sqrt{s}+\sqrt{u}}\leq|s-u|, ∣ u − s u ∣ = s + u u ∣ s − u ∣ ≤ ∣ s − u ∣ , so ∣ ( u − s u ) Γ d ( s − u ) ∣ ≤ 4 / π |(u-\sqrt{su})\Gamma_d(s-u)|\leq4/\pi ∣ ( u − s u ) Γ d ( s − u ) ∣ ≤ 4/ π s = u s=u s = u (58) , and division by s u ≥ t \sqrt{su}\geq t s u ≥ t (59) . ∎
The bounded remainder permits a direct comparison at growing offsets and removes the logarithmic loss from the observation-time covariance estimate. Identifying the bounded remainder would require a separate band-edge analysis.
An offset bound as a consequence
The following form is useful for the quantitative common-time comparisons. It follows by dividing the bilinear estimate by t t t
Theorem 6.5 (Growing-offset energy bound). For integers t ≥ 2 t \ge2 t ≥ 2 and 0 ≤ M ≤ t / 2 0 \le M \le t/2 0 ≤ M ≤ t /2
sup ∣ a ∣ , ∣ b ∣ ≤ M ∣ 2 t ∑ k ≥ 1 k a t + a , k a t + b , k − Γ d ( a − b ) ∣ ≤ C d 1 + M t . (60) \sup_{|a|,|b|\le M}\left\lvert\frac{2}{t}\sum_{k\ge1}ka_{t+a,k}a_{t+b,k}-\Gamma_d(a-b)\right\rvert\le C_d\frac{1+M}{t}. \tag*{(60)} ∣ a ∣ , ∣ b ∣ ≤ M sup t 2 k ≥ 1 ∑ k a t + a , k a t + b , k − Γ d ( a − b ) ≤ C d t 1 + M . (60) The supremum is over integer offsets. No average in t t t is taken. In particular, for each fixed pair of integer offsets,
2 t ∑ k ≥ 1 k a t + a , k a t + b , k ⟶ Γ d ( a − b ) , t → ∞ . \frac{2}{t}\sum_{k\ge1}ka_{t+a,k}a_{t+b,k}\longrightarrow\Gamma_d(a-b),\qquad t\to\infty. t 2 k ≥ 1 ∑ k a t + a , k a t + b , k ⟶ Γ d ( a − b ) , t → ∞. Proof. Apply Theorem 6.4 to s = t + a s=t+a s = t + a u = t + b u=t+b u = t + b t − M ≥ t / 2 t-M\ge t/2 t − M ≥ t /2
∣ ( t + a ) ( t + b ) t − 1 ∣ ≤ C M t , \left\lvert\frac{\sqrt{(t+a)(t+b)}}{t}-1\right\rvert\le C\frac{M}{t}, t ( t + a ) ( t + b ) − 1 ≤ C t M , and ∣ Γ d ( h ) ∣ ≤ c d < 2 \lvert\Gamma_d(h)\rvert\le c_d<2 ∣ Γ d ( h )∣ ≤ c d < 2 t t t
The ideal Poisson trace field
The field in (2.4) is built from independent Poisson variables. Interchanging the finite sums over divisors in (2.4) gives
P s = ∑ j = 3 s β s , j ( Z j − λ j ) , β s , j = 2 j ∑ m = 1 ⌊ s / j ⌋ a s , m j q − m j / 2 . (61) P_s=\sum_{j=3}^{s}\beta_{s,j}(Z_j-\lambda_j),\qquad\beta_{s,j}=2j\sum_{m=1}^{\lfloor s/j\rfloor}a_{s,mj}q^{-mj/2}. \tag*{(61)} P s = j = 3 ∑ s β s , j ( Z j − λ j ) , β s , j = 2 j m = 1 ∑ ⌊ s / j ⌋ a s , mj q − mj /2 . (61) Extend β s , j \beta_{s,j} β s , j j > s j>s j > s
∣ β s , j ∣ ≤ B j : = 2 2 j q − j / 2 1 − q − j / 2 , sup s ≥ 0 , j ≥ 3 ∣ β s , j ∣ ≤ C q . (62) \lvert\beta_{s,j}\rvert\le B_j:=\frac{2\sqrt{2}\,j q^{-j/2}}{1-q^{-j/2}},\qquad\sup_{\substack{s\ge0,\ j\ge3}}\lvert\beta_{s,j}\rvert\le C_q. \tag*{(62)} ∣ β s , j ∣ ≤ B j := 1 − q − j /2 2 2 j q − j /2 , s ≥ 0 , j ≥ 3 sup ∣ β s , j ∣ ≤ C q . (62) Period and covariance
For independent standard normals ξ j \xi_j ξ j P s G = ∑ j ≥ 3 β s , j λ j ξ j P_s^{\mathrm G}=\sum_{j\ge3}\beta_{s,j}\sqrt{\lambda_j}\xi_j P s G = ∑ j ≥ 3 β s , j λ j ξ j P s P_s P s
Lemma 7.1 (Non-bipartite and bipartite bases). The non-backtracking matrix of a finite connected simple d d d regular graph with d ≥ 3 d\ge3 d ≥ 3 is irreducible. Its period is p B p_B p B and it is primitive exactly when B B B is not bipartite. For some ϑ B ∈ ( 0 , 1 ) \vartheta_B\in(0,1) ϑ B ∈ ( 0 , 1 )
∣ b k q − k − η B ( k ) ∣ ≤ C B ϑ B k , η B ( k ) = 1 + ( p B − 1 ) ( − 1 ) k . (63) \left\lvert b_kq^{-k}-\eta_B(k)\right\rvert\le C_B\vartheta_B^k,\qquad\eta_B(k)=1+(p_B-1)(-1)^k. \tag*{(63)} b k q − k − η B ( k ) ≤ C B ϑ B k , η B ( k ) = 1 + ( p B − 1 ) ( − 1 ) k . (63) If B B B is bipartite, then, at every odd integer time s s s both A N 0 ( s ) A_N^0(s) A N 0 ( s ) and A N ∗ ( s ) A_N^*(s) A N ∗ ( s ) vanish identically, as do P s P_s P s and P s G P_s^{\mathrm G} P s G
Proof. The directed graph of allowed non-backtracking transitions has indegree and outdegree q q q K d , d K_{d,d} K d , d d ≥ 3 d \ge3 d ≥ 3 B B B
The Ihara–Bass characteristic identity is
det ( z I − H B ) = ( z 2 − 1 ) m 0 − n 0 det ( z 2 I − z A B + q I ) ; (64) \det(zI-H_B)=(z^2-1)^{m_0-n_0}\det(z^2I-zA_B+qI); \tag*{(64)} det ( z I − H B ) = ( z 2 − 1 ) m 0 − n 0 det ( z 2 I − z A B + q I ) ; (64) see [3 ] , or the equivalent eigenvalue statement in [14 Lemma 5.3] . If ∣ z ∣ = q > 1 |z|=q>1 ∣ z ∣ = q > 1 z + q / z z+q/z z + q / z z = q e i θ z=qe^{i\theta} z = q e i θ ( q − 1 ) sin θ (q-1)\sin\theta ( q − 1 ) sin θ z = ± q z=\pm q z = ± q d d d B B B − d -d − d B B B
x T ( d I − A B ) x = ∑ { v , w } ∈ E ( x v − x w ) 2 , x T ( d I + A B ) x = ∑ { v , w } ∈ E ( x v + x w ) 2 . x^{\mathsf T}(dI-A_B)x=\sum_{\{v,w\}\in E}(x_v-x_w)^2,\qquad x^{\mathsf T}(dI+A_B)x=\sum_{\{v,w\}\in E}(x_v+x_w)^2. x T ( d I − A B ) x = { v , w } ∈ E ∑ ( x v − x w ) 2 , x T ( d I + A B ) x = { v , w } ∈ E ∑ ( x v + x w ) 2 . In the second identity a nonzero kernel vector changes sign along every edge, which is possible exactly for a bipartition. The root q q q (64) , and the root − q -q − q q > 1 q>1 q > 1 q q q (63) , whether or not the matrix is diagonalisable.
A lift of a bipartite graph is bipartite. Assign to a directed edge the sign of its initial vertex in the lifted bipartition, and let J J J U G N U_{G_N} U G N J U G N J = − U G N JU_{G_N}J=-U_{G_N} J U G N J = − U G N b j = C N , j = 0 b_j=C_{N,j}=0 b j = C N , j = 0 j j j D N ( s ) = m N ( s ) = 0 D_N(s)=m_N(s)=0 D N ( s ) = m N ( s ) = 0 s s s P s P_s P s (61) has only zero contributions at odd s s s
Lemma 7.2 (Covariance correction and linear variance bound). For every base B B B η B \eta_B η B (63) ,
∑ k , ℓ ≥ 1 ∣ Λ B ( k , ℓ ) − 2 k η B ( k ) 1 { k = ℓ } ∣ < ∞ . (65) \sum_{k,\ell\ge1}\left|\Lambda_B(k,\ell)-2k\eta_B(k)\mathbf{1}_{\{k=\ell\}}\right|<\infty. \tag*{(65)} k , ℓ ≥ 1 ∑ Λ B ( k , ℓ ) − 2 k η B ( k ) 1 { k = ℓ } < ∞. (65) Uniformly for nonnegative integer s , u s,u s , u
∣ Cov ( P s , P u ) − 2 ∑ k ≥ 1 k η B ( k ) a s , k a u , k ∣ ≤ C B , Var ( P s ) ≤ 8 m 0 s + C B . (66) \left|\operatorname{Cov}(P_s,P_u)-2\sum_{k\ge1}k\eta_B(k)a_{s,k}a_{u,k}\right|\le C_B,\qquad\operatorname{Var}(P_s)\le8m_0s+C_B. \tag*{(66)} Cov ( P s , P u ) − 2 k ≥ 1 ∑ k η B ( k ) a s , k a u , k ≤ C B , Var ( P s ) ≤ 8 m 0 s + C B . (66) For a non-bipartite base, η B ( k ) = 1 \eta_B(k)=1 η B ( k ) = 1
Proof. Set
d k = 2 k ( b k q − k − η B ( k ) ) , E k ℓ = 2 q − ( k + ℓ ) / 2 ∑ j ∣ gcd ( k , ℓ ) , j > 3 ( k , ℓ ) ≠ ( j , j ) j b j ≥ 0. d_k=2k(b_kq^{-k}-\eta_B(k)),\qquad E_{k\ell}=2q^{-(k+\ell)/2}\sum_{\substack{j\mid\gcd(k,\ell),\ j>3\\(k,\ell)\ne(j,j)}}jb_j\ge0. d k = 2 k ( b k q − k − η B ( k )) , E k ℓ = 2 q − ( k + ℓ ) /2 j ∣ g c d ( k , ℓ ) , j > 3 ( k , ℓ ) = ( j , j ) ∑ j b j ≥ 0. Equation (2.1) gives
Λ B ( k , ℓ ) − 2 k η B ( k ) 1 { k = ℓ } = d k 1 { k = ℓ } + E k ℓ . (67) \Lambda_B(k,\ell)-2k\eta_B(k)\mathbf{1}_{\{k=\ell\}}=d_k\mathbf{1}_{\{k=\ell\}}+E_{k\ell}. \tag*{(67)} Λ B ( k , ℓ ) − 2 k η B ( k ) 1 { k = ℓ } = d k 1 { k = ℓ } + E k ℓ . (67) The diagonal correction is absolutely summable by (7.3). For z j = q − j / 2 z_j=q^{-j/2} z j = q − j /2 b j q − j ≤ 2 m 0 b_jq^{-j}\le2m_0 b j q − j ≤ 2 m 0
∑ k , ℓ ≥ 1 E k ℓ = 2 ∑ j ≥ 3 j b j { ( z j 1 − z j ) 2 − z j 2 } = 2 ∑ j ≥ 3 j ( b j q − j ) z j 2 ( 2 − z j ) ( 1 − z j ) 2 ≤ C B ∑ j ≥ 3 j q − j / 2 < ∞ . \sum_{k,\ell\ge1} E_{k\ell}=2\sum_{j\ge3}jb_j\left\{\left(\frac{z_j}{1-z_j}\right)^2-z_j^2\right\}
=2\sum_{j\ge3}j(b_jq^{-j})\frac{z_j^2(2-z_j)}{(1-z_j)^2}\le C_B\sum_{j\ge3}jq^{-j/2}<\infty. k , ℓ ≥ 1 ∑ E k ℓ = 2 j ≥ 3 ∑ j b j { ( 1 − z j z j ) 2 − z j 2 } = 2 j ≥ 3 ∑ j ( b j q − j ) ( 1 − z j ) 2 z j 2 ( 2 − z j ) ≤ C B j ≥ 3 ∑ j q − j /2 < ∞. Since all summands are nonnegative, Tonelli’s theorem justifies the rearrangement. The triangle inequality proves (7.5); multiplication by the bounded coefficients gives the covariance assertion. For the variance, apply b k ≤ 2 m 0 q k b_k\le2m_0q^k b k ≤ 2 m 0 q k
Var ( P s ) ≤ 4 m 0 ∑ k k a s , k 2 + 2 ∑ k , ℓ E k ℓ ≤ 8 m 0 s + C B . \operatorname{Var}(P_s)\le4m_0\sum_kka_{s,k}^2+2\sum_{k,\ell}E_{k\ell}\le8m_0s+C_B. Var ( P s ) ≤ 4 m 0 k ∑ k a s , k 2 + 2 k , ℓ ∑ E k ℓ ≤ 8 m 0 s + C B . Proposition 7.3 (Covariance with deterministic time growth). Put
J B ( s , u ) = p B 1 { p B ∣ s , p B ∣ u } Γ d ( s − u ) . J_B(s,u)=p_B\mathbf{1}_{\{p_B\mid s,\ p_B\mid u\}}\Gamma_d(s-u). J B ( s , u ) = p B 1 { p B ∣ s , p B ∣ u } Γ d ( s − u ) . For all positive integers s , u s,u s , u
∣ Cov ( P s , P u ) − s u J B ( s , u ) ∣ ≤ C B . (68) \left|\operatorname{Cov}(P_s,P_u)-\sqrt{su}J_B(s,u)\right|\le C_B. \tag*{(68)} Cov ( P s , P u ) − s u J B ( s , u ) ≤ C B . (68) Consequently, for all positive integers s , u s,u s , u
∣ Cov ( P s , P u ) s u − J B ( s , u ) ∣ ≤ C B s u . (69) \left|\frac{\operatorname{Cov}(P_s,P_u)}{\sqrt{su}}-J_B(s,u)\right|\le\frac{C_B}{\sqrt{su}}. \tag*{(69)} s u Cov ( P s , P u ) − J B ( s , u ) ≤ s u C B . (69) In particular, uniformly for s , u ∈ [ t , 2 t ] s,u\in[t,2t] s , u ∈ [ t , 2 t ]
∣ Cov ( P s , P u ) s u − J B ( s , u ) ∣ ≤ C B / t . (70) \left|\frac{\operatorname{Cov}(P_s,P_u)}{\sqrt{su}}-J_B(s,u)\right|\le C_B/t. \tag*{(70)} s u Cov ( P s , P u ) − J B ( s , u ) ≤ C B / t . (70) If R = max ( s , u ) R=\max(s,u) R = max ( s , u ) s , u ∈ [ t , 2 t ] s,u\in[t,2t] s , u ∈ [ t , 2 t ] N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
∣ Cov ( Tr U G N s , Tr U G N u ) − s u J B ( s , u ) ∣ ≤ C B { 1 + R 6 q R N } . (71) \left|\operatorname{Cov}(\operatorname{Tr}U_{G_N}^s,\operatorname{Tr}U_{G_N}^u)-\sqrt{su}J_B(s,u)\right|
\le C_B\left\{1+\frac{R^6q^R}{N}\right\}. \tag*{(71)} Cov ( Tr U G N s , Tr U G N u ) − s u J B ( s , u ) ≤ C B { 1 + N R 6 q R } . (71) The error after division by s u \sqrt{su} s u C B { t − 1 + R 5 q R / N } C_B\{t^{-1}+R^5q^R/N\} C B { t − 1 + R 5 q R / N }
Proof. For a non-bipartite base, Lemma 7.2 and Theorem 6.4 apply directly. For a bipartite base and even s , u s,u s , u
2 ∑ k k η B ( k ) a s , k a u , k = 4 ∑ k k a s , k a u , k , 2\sum_k k\eta_B(k)a_{s,k}a_{u,k}=4\sum_kka_{s,k}a_{u,k}, 2 k ∑ k η B ( k ) a s , k a u , k = 4 k ∑ k a s , k a u , k , which is twice the non-bipartite energy; this agrees with p B = 2 p_B=2 p B = 2 J B ( s , u ) J_B(s,u) J B ( s , u )
For the finite lift use the exact representation (2.3) and Proposition 5.6. With S a ( s ) = ∑ k = 3 ∞ ∣ a s , k ∣ k q k / 2 S_a(s)=\sum_{k=3}^{\infty}|a_{s,k}|kq^{k/2} S a ( s ) = ∑ k = 3 ∞ ∣ a s , k ∣ k q k /2
∣ Cov ( A N ( s ) , A N ( u ) ) − Cov ( P s , P u ) ∣ ≤ C B N { S 2 ( s ) S 2 ( u ) + S 1 ( s ) S 5 ( u ) + S 5 ( s ) S 1 ( u ) + S 3 ( s ) S 3 ( u ) } ≤ C B R 6 q R / N . (72) \left|\operatorname{Cov}(\mathcal{A}_N(s),\mathcal{A}_N(u))-\operatorname{Cov}(P_s,P_u)\right|
\le\frac{C_B}{N}\left\{S_2(s)S_2(u)+S_1(s)S_5(u)+S_5(s)S_1(u)+S_3(s)S_3(u)\right\}
\le C_BR^6q^R/N. \tag*{(72)} ∣ Cov ( A N ( s ) , A N ( u )) − Cov ( P s , P u ) ∣ ≤ N C B { S 2 ( s ) S 2 ( u ) + S 1 ( s ) S 5 ( u ) + S 5 ( s ) S 1 ( u ) + S 3 ( s ) S 3 ( u ) } ≤ C B R 6 q R / N . (72) This comparison uses only s , u ≤ R s,u\le R s , u ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 s u ≥ t \sqrt{su}\ge t s u ≥ t R / t ≤ 2 R/t\le2 R / t ≤ 2
For the next lemma, put T M = ( Γ d ( a − b ) ) 0 ≤ a , b < M T_M = (\Gamma_d(a-b))_{0\leq a,b<M} T M = ( Γ d ( a − b ) ) 0 ≤ a , b < M ∥ ⋅ ∥ o p \lVert\cdot\rVert_{\mathrm{op}} ∥ ⋅ ∥ op
Lemma 7.4 (Constants for the Grover covariance). For γ = Γ d \gamma= \Gamma_d γ = Γ d
sup M ∥ T M ∥ o p ≤ 2 , ∑ h ∈ Z Γ d ( h ) 2 = 2 c d . \sup_M \lVert T_M\rVert_{\mathrm{op}} \leq2,\qquad\sum_{h\in\mathbb{Z}} \Gamma_d(h)^2 = 2c_d. M sup ∥ T M ∥ op ≤ 2 , h ∈ Z ∑ Γ d ( h ) 2 = 2 c d . Proof. Let I d = [ α d , π − α d ] ∪ [ − π + α d , − α d ] I_d=[\alpha_d,\pi-\alpha_d]\cup[-\pi+\alpha_d,-\alpha_d] I d = [ α d , π − α d ] ∪ [ − π + α d , − α d ]
Γ d ( h ) = 1 π ∫ I d e i h θ d θ . \Gamma_d(h)=\frac{1}{\pi}\int_{I_d}e^{ih\theta}\,\mathrm{d}\theta. Γ d ( h ) = π 1 ∫ I d e ih θ d θ . For every finite complex vector z z z
z ∗ T M z = 1 π ∫ I d ∣ ∑ a = 0 M − 1 z a e i a θ ∣ 2 d θ ≤ 2 ∑ a ∣ z a ∣ 2 . z^*T_Mz=\frac{1}{\pi}\int_{I_d}\left|\sum_{a=0}^{M-1}z_a e^{ia\theta}\right|^2\,\mathrm{d}\theta\leq2\sum_a|z_a|^2. z ∗ T M z = π 1 ∫ I d a = 0 ∑ M − 1 z a e ia θ 2 d θ ≤ 2 a ∑ ∣ z a ∣ 2 . This also proves positivity. In the normalised Fourier convention, Γ d ( h ) \Gamma_d(h) Γ d ( h ) 21 I d 2\mathbf{1}_{I_d} 2 1 I d
∑ h Γ d ( h ) 2 = 1 2 π ∫ − π π 41 I d ( θ ) d θ = 4 ( π − 2 α d ) π = 2 c d . \sum_h\Gamma_d(h)^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}4\mathbf{1}_{I_d}(\theta)\,\mathrm{d}\theta=\frac{4(\pi-2\alpha_d)}{\pi}=2c_d. h ∑ Γ d ( h ) 2 = 2 π 1 ∫ − π π 4 1 I d ( θ ) d θ = π 4 ( π − 2 α d ) = 2 c d . Fixed-offset fluctuations
For a non-bipartite base, Theorem 6.5 and Lemma 7.2 give
1 t Cov ( P t + a , P t + b ) ⟶ Γ d ( a − b ) for fixed integer a , b . (73) \frac{1}{t}\operatorname{Cov}(P_{t+a},P_{t+b})\longrightarrow\Gamma_d(a-b)\quad\text{for fixed integer }a,b. \tag*{(73)} t 1 Cov ( P t + a , P t + b ) ⟶ Γ d ( a − b ) for fixed integer a , b . (73) Proposition 7.5 (Gaussian limit for the Poisson field). Under primitivity, for any fixed offsets h 1 , … , h p h_1,\ldots,h_p h 1 , … , h p
( P t + h a / t ) a = 1 p → d ( G h a ) a = 1 p . \left(P_{t+h_a}/\sqrt{t}\right)_{a=1}^{p}\xrightarrow{\mathrm{d}}\left(G_{h_a}\right)_{a=1}^{p}. ( P t + h a / t ) a = 1 p d ( G h a ) a = 1 p . The covariance entries converge, and convergence also holds in W 2 \mathrm{W}_2 W 2 ( Y k ) k = 3 L (Y_k)_{k=3}^{L} ( Y k ) k = 3 L
Proof. In subscripts on constants, h h h ( h 1 , … , h p ) (h_1,\ldots,h_p) ( h 1 , … , h p ) c = ( c 1 , … , c p ) ∈ R p c=(c_1,\ldots,c_p)\in\mathbb{R}^p c = ( c 1 , … , c p ) ∈ R p u t , j = t − 1 / 2 ∑ a c a β t + h a , j u_{t,j}=t^{-1/2}\sum_a c_a\beta_{t+h_a,j} u t , j = t − 1/2 ∑ a c a β t + h a , j (62) , max j ∣ u t , j ∣ ≤ C q , c t − 1 / 2 \max_j|u_{t,j}|\leq C_{q,c}t^{-1/2} max j ∣ u t , j ∣ ≤ C q , c t − 1/2
v t = ∑ j λ j u t , j 2 v_t=\sum_j\lambda_j u_{t,j}^2 v t = j ∑ λ j u t , j 2 is bounded and, by (73) , tends to ∑ a , b c a c b Γ d ( h a − h b ) \sum_{a,b}c_ac_b\Gamma_d(h_a-h_b) ∑ a , b c a c b Γ d ( h a − h b )
log E [ exp ( i z ∑ j u t , j ( Z j − λ j ) ) ] = ∑ j λ j ( e i z u t , j − 1 − i z u t , j ) , \log\mathbb{E}\left[\exp\left(iz\sum_j u_{t,j}(Z_j-\lambda_j)\right)\right]
=\sum_j\lambda_j\left(e^{izu_{t,j}}-1-izu_{t,j}\right), log E [ exp ( i z j ∑ u t , j ( Z j − λ j ) ) ] = j ∑ λ j ( e i z u t , j − 1 − i z u t , j ) , where the right-hand side specifies the continuous branch of the logarithm that vanishes at z = 0 z=0 z = 0 − z 2 v t / 2 -z^2v_t/2 − z 2 v t /2
∣ z ∣ 3 6 ∑ j λ j ∣ u t , j ∣ 3 ≤ ∣ z ∣ 3 6 max j ∣ u t , j ∣ v t ≤ C B , c , h ∣ z ∣ 3 t − 1 / 2 . \frac{|z|^3}{6}\sum_j\lambda_j|u_{t,j}|^3\leq\frac{|z|^3}{6}\max_j|u_{t,j}|v_t\leq C_{B,c,h}|z|^3t^{-1/2}. 6 ∣ z ∣ 3 j ∑ λ j ∣ u t , j ∣ 3 ≤ 6 ∣ z ∣ 3 j max ∣ u t , j ∣ v t ≤ C B , c , h ∣ z ∣ 3 t − 1/2 . The characteristic-function argument and the Cramér–Wold device prove the Gaussian limit. Poisson variables with zero mean vanish identically.
To prove independence from the fixed low-mode vector, delete the terms 3 ≤ j ≤ L 3 \le j \le L 3 ≤ j ≤ L (61) . Their normalised L 2 L^2 L 2 O B , L ( t − 1 / 2 ) O_{B,L}(t^{-1/2}) O B , L ( t − 1/2 ) (62) . The remaining vector depends only on the independent variables Z j Z_j Z j j > L j>L j > L ( Y k ) k = 3 L (Y_k)_{k=3}^L ( Y k ) k = 3 L (73) gives convergence of the expected squared Euclidean norms of the vectors at growing times; the low-mode vector has a fixed finite second moment. Weak convergence and convergence of second moments yield W 2 W_2 W 2
We verify nondegeneracy and then give an explicit representation of the limiting field. For distinct offsets and a nonzero real vector c c c
∑ a , b c a c b Γ d ( h a − h b ) = 2 π ∫ α d π − α d ∣ ∑ a c a e i h a θ ∣ 2 d θ > 0. \sum_{a,b} c_a c_b \Gamma_d(h_a-h_b) = \frac{2}{\pi} \int_{\alpha_d}^{\pi-\alpha_d} \left|\sum_a c_a e^{i h_a \theta}\right|^2 \,d\theta> 0. a , b ∑ c a c b Γ d ( h a − h b ) = π 2 ∫ α d π − α d a ∑ c a e i h a θ 2 d θ > 0. A nonzero trigonometric polynomial cannot vanish on this whole interval. Equivalently, with independent real Gaussian white-noise measures W 1 W_1 W 1 W 2 W_2 W 2
G h = 2 π ∫ α d π − α d { cos ( h θ ) W 1 ( d θ ) + sin ( h θ ) W 2 ( d θ ) } . G_h = \sqrt{\frac{2}{\pi}} \int_{\alpha_d}^{\pi-\alpha_d} \{\cos(h\theta) W_1(d\theta) + \sin(h\theta) W_2(d\theta)\}. G h = π 2 ∫ α d π − α d { cos ( h θ ) W 1 ( d θ ) + sin ( h θ ) W 2 ( d θ )} . This defines a stationary real Gaussian process indexed by the integers, with covariance (1) .
Transfer to growing times on the finite lift
The distributional and second-moment transfers are treated separately, using the distance conventions from § 1 .
Proposition 8.1 (Finite-lift distributional comparison). Fix offsets h 1 , … , h p h_1,\ldots,h_p h 1 , … , h p H = max a ∣ h a ∣ H=\max_a |h_a| H = max a ∣ h a ∣ For integers t ≥ 2 H + 3 t \ge2H+3 t ≥ 2 H + 3 put R = t + H R=t+H R = t + H If N ≥ 8 R 2 N \ge8R^2 N ≥ 8 R 2 then
d B L ( L ( ( A N ( t + h a ) / t ) a ) , L ( ( P t + h a / t ) a ) ) ≤ C B , h R 4 q R N . (74) d_{\mathrm{BL}}\left(\mathcal{L}\left((A_N(t+h_a)/\sqrt{t})_a\right),\mathcal{L}\left((P_{t+h_a}/\sqrt{t})_a\right)\right) \le C_{B,h}\frac{R^4q^R}{N}. \tag*{(74)} d BL ( L ( ( A N ( t + h a ) / t ) a ) , L ( ( P t + h a / t ) a ) ) ≤ C B , h N R 4 q R . (74) The same estimate, with a constant also depending on fixed L L L ( Y N , k ) k = 3 L (Y_{N,k})_{k=3}^L ( Y N , k ) k = 3 L ( Y k ) k = 3 L (Y_k)_{k=3}^L ( Y k ) k = 3 L t t t
Proof. Substitute (38) into (13) and apply Lemma 5.1. The normalised total L 1 L^1 L 1 s ≤ R s \le R s ≤ R
C B N t ∑ k = 3 s ∣ a s , k ∣ k 4 q k / 2 ≤ C B R 4 q R / 2 N t . \frac{C_B}{N\sqrt{t}}\sum_{k=3}^{s}|a_{s,k}|k^4q^{k/2} \le C_B\frac{R^4q^{R/2}}{N\sqrt{t}}. N t C B k = 3 ∑ s ∣ a s , k ∣ k 4 q k /2 ≤ C B N t R 4 q R /2 . Here (50) was used. The cycle-count expression, still centred at μ N , j \mu_{N,j} μ N , j R R R C B R 4 q R / N C_B R^4q^R/N C B R 4 q R / N λ j \lambda_j λ j
For a single unnormalised Chebyshev mode, Proposition 3.1 bounds the change of centre by
2 q − k / 2 ∑ j ∣ k , j ≥ 3 j ∣ μ N , j − λ j ∣ ≤ C B k 2 q k / 2 / N . (75) 2q^{-k/2}\sum_{\substack{j\mid k,\\ j\ge3}}j|\mu_{N,j}-\lambda_j| \le C_B k^2q^{k/2}/N. \tag*{(75)} 2 q − k /2 j ∣ k , j ≥ 3 ∑ j ∣ μ N , j − λ j ∣ ≤ C B k 2 q k /2 / N . (75) After summing with a s , k a_{s,k} a s , k t \sqrt{t} t C B R 2 q R / 2 / ( N t ) C_B R^2 q^{R/2}/(N\sqrt{t}) C B R 2 q R /2 / ( N t ) R / t R/t R / t q ≥ 2 q \ge2 q ≥ 2 (74) . Appending a fixed low-mode vector adds only C B , L / N C_{B,L}/N C B , L / N
Proposition 8.2 (Finite-lift covariance comparison). Fix integers a , b a,b a , b H = max { ∣ a ∣ , ∣ b ∣ } H=\max\{|a|,|b|\} H = max { ∣ a ∣ , ∣ b ∣ } t ≥ 2 H + 3 t\ge2H+3 t ≥ 2 H + 3 R = t + H R=t+H R = t + H N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
∣ 1 t Cov ( A N ( t + a ) , A N ( t + b ) ) − 1 t Cov ( P t + a , P t + b ) ∣ ≤ C B , a , b R 5 q R N . \left|\frac{1}{t}\operatorname{Cov}(A_N(t+a),A_N(t+b))-\frac{1}{t}\operatorname{Cov}(P_{t+a},P_{t+b})\right| \le C_{B,a,b}\frac{R^5q^R}{N}. t 1 Cov ( A N ( t + a ) , A N ( t + b )) − t 1 Cov ( P t + a , P t + b ) ≤ C B , a , b N R 5 q R . Primitivity is not required for this comparison.
Proof. The weighted finite–ideal estimate (72) uses only s , u ≤ R s,u\le R s , u ≤ R N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 s = t + a s=t+a s = t + a u = t + b u=t+b u = t + b t t t R / t ≤ 3 / 2 R/t\le3/2 R / t ≤ 3/2 C B R 6 q R / ( N t ) ≤ C B , a , b R 5 q R / N C_B R^6q^R/(Nt)\le C_{B,a,b}R^5q^R/N C B R 6 q R / ( N t ) ≤ C B , a , b R 5 q R / N
For the remainder of this section, suppose that the base is non-bipartite. For fixed offsets, define the deterministic quantity
ϵ r d , a , b ( t ) = ∣ 2 t ∑ k ≥ 1 k a t + a , k a t + b , k − Γ d ( a − b ) ∣ . \epsilon_{rd,a,b}(t)=\left|\frac{2}{t}\sum_{k\ge1}ka_{t+a,k}a_{t+b,k}-\Gamma_d(a-b)\right|. ϵ r d , a , b ( t ) = t 2 k ≥ 1 ∑ k a t + a , k a t + b , k − Γ d ( a − b ) . Theorem 6.5 proves ϵ r d , a , b ( t ) → 0 \epsilon_{rd,a,b}(t)\to0 ϵ r d , a , b ( t ) → 0
∣ 1 t Cov ( A N ( t + a ) , A N ( t + b ) ) − Γ d ( a − b ) ∣ ≤ C B , a , b ( R 5 q R N + 1 t ) + ϵ r d , a , b ( t ) . (76) \left|\frac{1}{t}\operatorname{Cov}(A_N(t+a),A_N(t+b))-\Gamma_d(a-b)\right|\le C_{B,a,b}\left(\frac{R^5q^R}{N}+\frac{1}{t}\right)+\epsilon_{rd,a,b}(t). \tag*{(76)} t 1 Cov ( A N ( t + a ) , A N ( t + b )) − Γ d ( a − b ) ≤ C B , a , b ( N R 5 q R + t 1 ) + ϵ r d , a , b ( t ) . (76) Theorem 6.5 gives the explicit bound ϵ r d , a , b ( t ) ≤ C d ( 1 + max { ∣ a ∣ , ∣ b ∣ } ) / t \epsilon_{rd,a,b}(t)\le C_d(1+\max\{|a|,|b|\})/t ϵ r d , a , b ( t ) ≤ C d ( 1 + max { ∣ a ∣ , ∣ b ∣ }) / t t t t
Proof of Theorem 1.5 and Corollary 1.6. Condition (6) ensures N ≥ 8 R N 2 N\ge8R_N^2 N ≥ 8 R N 2 Proposition 8.1 and Proposition 7.5 give (7) , including joint convergence with the independent fixed low-mode vector. Under (8) , which implies (6) , equation (8.2) gives convergence of all covariances of the high-time vectors. Their coordinates have mean zero, so their expected squared Euclidean norms converge to that of the Gaussian limit. The second moments of the fixed low-mode vector converge by (48) . Consequently, the full joint vector converges weakly and its second moments converge.
To see that this suffices for W 2 W_2 W 2 ∣ x − y ∣ = x + y − 2 min ( x , y ) |x-y|=x+y-2\min(x,y) ∣ x − y ∣ = x + y − 2 min ( x , y ) L 1 L^1 L 1 W 2 W_2 W 2
Taking a = b = 0 a=b=0 a = b = 0 (9) . Formula (10) follows by direct integration of (1) .
Explicit centring and quantitative time averages
We first give the centring and a transfer estimate valid for both periods and both time normalisations. We then establish finite-size bounds for common-time averages on a non-bipartite base. The observation-time law of large numbers is proved in § 10.7 .
An explicit centring from the base
The following centring is computed directly from the base graph and the number of sheets. Recall β s , j \beta_{s,j} β s , j (61) and put
L N ( s ) = ∑ k = 3 s a s , k q − k / 2 R N , k . L_N(s)=\sum_{k=3}^{s}a_{s,k}q^{-k/2}\mathcal{R}_{N,k}. L N ( s ) = k = 3 ∑ s a s , k q − k /2 R N , k . Define
D N ( s ) = N ( m 0 − n 0 ) ( 1 + ( − 1 ) s ) + 2 n 0 N a s , 0 − N ( m 0 − n 0 ) ∑ k = 1 s a s , k q − k / 2 ( 1 + ( − 1 ) k ) , D_N(s)=N(m_0-n_0)(1+(-1)^s)+2n_0Na_{s,0}
-N(m_0-n_0)\sum_{k=1}^{s}a_{s,k}q^{-k/2}(1+(-1)^k), D N ( s ) = N ( m 0 − n 0 ) ( 1 + ( − 1 ) s ) + 2 n 0 N a s , 0 − N ( m 0 − n 0 ) k = 1 ∑ s a s , k q − k /2 ( 1 + ( − 1 ) k ) , w k ∗ = ∑ j ∣ k , j ≥ 3 b j , m N ( s ) = D N ( s ) + ∑ k = 1 s a s , k q − k / 2 w k ∗ . w_k^*=\sum_{\substack{j\mid k,\,j\geq3}}b_j,\qquad
m_N(s)=D_N(s)+\sum_{k=1}^{s}a_{s,k}q^{-k/2}w_k^*. w k ∗ = j ∣ k , j ≥ 3 ∑ b j , m N ( s ) = D N ( s ) + k = 1 ∑ s a s , k q − k /2 w k ∗ . These are finite sums computable from B B B N N N s s s s = 0 s=0 s = 0 m N ( 0 ) = 2 m 0 N m_N(0)=2m_0N m N ( 0 ) = 2 m 0 N
Proposition 9.1 (Deterministic and exact mean centrings). For all s ≥ 0 s\geq0 s ≥ 0
Tr U G N s = D N ( s ) + ∑ k = 1 s a s , k q − k / 2 W N , k . (77) \operatorname{Tr}U_{G_N}^s=D_N(s)+\sum_{k=1}^{s}a_{s,k}q^{-k/2}W_{N,k}. \tag*{(77)} Tr U G N s = D N ( s ) + k = 1 ∑ s a s , k q − k /2 W N , k . (77) In particular,
Tr U G N s − m N ( s ) = ∑ j = 3 s β s , j ( C N , j − λ j ) + L N ( s ) . (78) \operatorname{Tr}U_{G_N}^s-m_N(s)=\sum_{j=3}^{s}\beta_{s,j}(C_{N,j}-\lambda_j)+L_N(s). \tag*{(78)} Tr U G N s − m N ( s ) = j = 3 ∑ s β s , j ( C N , j − λ j ) + L N ( s ) . (78) For s ≤ R s\leq R s ≤ R N ≥ 8 R 2 N\geq8R^2 N ≥ 8 R 2
∣ E [ Tr U G N s ] − m N ( s ) ∣ ≤ C B R 4 q R / 2 N . (79) \left|\mathbb{E}\left[\operatorname{Tr}U_{G_N}^s\right]-m_N(s)\right|
\leq C_B\frac{R^4q^{R/2}}{N}. \tag*{(79)} E [ Tr U G N s ] − m N ( s ) ≤ C B N R 4 q R /2 . (79) Proof. For a regular graph with n n n m m m
Tr U G s = ( m − n ) ( 1 + ( − 1 ) s ) + 2 Tr T s ( A G / d ) , \operatorname{Tr}U_G^s=(m-n)(1+(-1)^s)+2\operatorname{Tr}T_s(A_G/d), Tr U G s = ( m − n ) ( 1 + ( − 1 ) s ) + 2 Tr T s ( A G / d ) , 2 Tr T k ( A G / ( 2 q ) ) = q − k / 2 { Tr H k G − ( m − n ) ( 1 + ( − 1 ) k ) } , k ≥ 1. 2\operatorname{Tr}T_k(A_G/(2\sqrt{q}))=q^{-k/2}\left\{\operatorname{Tr}H_k^G-(m-n)(1+(-1)^k)\right\},\qquad k\geq1. 2 Tr T k ( A G / ( 2 q )) = q − k /2 { Tr H k G − ( m − n ) ( 1 + ( − 1 ) k ) } , k ≥ 1. They hold componentwise for disconnected graphs. Substitute (12) and n = n 0 N n=n_0N n = n 0 N m = m 0 N m=m_0N m = m 0 N 2 n 0 N a s , 0 2n_0Na_{s,0} 2 n 0 N a s , 0 (77) . Since w k ∗ = ∑ j ∣ k 2 j λ j w_k^*=\sum_{j\mid k}2j\lambda_j w k ∗ = ∑ j ∣ k 2 j λ j (35) gives (78) .
The difference between the exact mean and the explicit centring is bounded by
∑ k = 3 s ∣ a s , k ∣ q − k / 2 { ∑ j ∣ k , j ≥ 3 2 j ∣ μ N , j − λ j ∣ + E [ R N , k ] } . \sum_{k=3}^{s}|a_{s,k}|q^{-k/2}\left\{\sum_{\substack{j\mid k,\,j\geq3}}2j|\mu_{N,j}-\lambda_j|+\mathbb{E}[\mathcal{R}_{N,k}]\right\}. k = 3 ∑ s ∣ a s , k ∣ q − k /2 ⎩ ⎨ ⎧ j ∣ k , j ≥ 3 ∑ 2 j ∣ μ N , j − λ j ∣ + E [ R N , k ] ⎭ ⎬ ⎫ . By Proposition 3.1, the divisor sum is at most C B k 2 q k / N C_Bk^2q^k/N C B k 2 q k / N C B k 4 q k / N C_Bk^4q^k/N C B k 4 q k / N p = 4 p=4 p = 4 □ \square □
The centring is thus an explicit finite sum. For numerical work, the coefficients can be evaluated by recurrence, and the new-sector compression in the first paper [1 ] can be used to compute the fluctuating traces without numerically subtracting inherited eigenvalues. Its numerical conditioning and floating-point stability require separate analysis for large times or sheet numbers.
A common transfer estimate
For t ≥ 4 t \ge4 t ≥ 4 1 ≤ M ≤ t / 2 1 \le M \le t/2 1 ≤ M ≤ t /2 R = t + M R=t+M R = t + M N ≥ 8 R 2 N \ge8R^2 N ≥ 8 R 2 C N = ( C N , j ) 3 ≤ j ≤ R C_N=(C_{N,j})_{3\le j\le R} C N = ( C N , j ) 3 ≤ j ≤ R
X N , t , h = t − 1 / 2 ∑ j = 3 R β t + h , j ( C N , j − λ j ) , Q N , t , M c y c = M − 1 ∑ h < M X N , t , h 2 . \mathcal{X}_{N,t,h}=t^{-1/2}\sum_{j=3}^{R}\beta_{t+h,j}(C_{N,j}-\lambda_j), \qquad Q^{\mathrm{cyc}}_{N,t,M}=M^{-1}\sum_{h<M}\mathcal{X}_{N,t,h}^2. X N , t , h = t − 1/2 j = 3 ∑ R β t + h , j ( C N , j − λ j ) , Q N , t , M cyc = M − 1 h < M ∑ X N , t , h 2 . For 3 ≤ k ≤ R 3 \le k \le R 3 ≤ k ≤ R Ψ k ( C N ) = 2 q − k / 2 ∑ j ∣ k , j ≥ 3 j ( C N , j − λ j ) \Psi_k(C_N)=2q^{-k/2}\sum_{j\mid k,\ j\ge3}j(C_{N,j}-\lambda_j) Ψ k ( C N ) = 2 q − k /2 ∑ j ∣ k , j ≥ 3 j ( C N , j − λ j ) C N C_N C N ( Z j ) 3 ≤ j ≤ R (Z_j)_{3\le j\le R} ( Z j ) 3 ≤ j ≤ R C B τ N , R C_B\tau_{N,R} C B τ N , R Ψ k \Psi_k Ψ k L L L R ≥ L R \ge L R ≥ L
E [ ∑ k = 3 L ∣ Y N , k − Ψ k ( C N ) ∣ ] ≤ C B , L / N . (80) \mathbb{E}\left[\sum_{k=3}^{L}|Y_{N,k}-\Psi_k(C_N)|\right]\le C_{B,L}/N. \tag*{(80)} E [ k = 3 ∑ L ∣ Y N , k − Ψ k ( C N ) ∣ ] ≤ C B , L / N . (80) Indeed, (5.4) separates each difference into the centred remainder and the change from μ N , j \mu_{N,j} μ N , j λ j \lambda_j λ j L L L
Lemma 9.2 (Weighted transfer of squared traces). Let V h = A N ⋆ ( t + h ) / t V_h=\mathcal{A}^{\star}_{N}(t+h)/\sqrt{t} V h = A N ⋆ ( t + h ) / t X h = X N , t , h X_h=\mathcal{X}_{N,t,h} X h = X N , t , h E N ⋆ = max h < M ∣ V h − X h ∣ E_N^\star=\max_{h<M}|V_h-X_h| E N ⋆ = max h < M ∣ V h − X h ∣
E [ E N ⋆ ] ≤ C B κ N , t , M , κ N , t , M = M R 4 q R / 2 N t . (81) \mathbb{E}[E_N^\star]\le C_B\kappa_{N,t,M}, \qquad\kappa_{N,t,M}=\frac{MR^4q^{R/2}}{N\sqrt{t}}. \tag*{(81)} E [ E N ⋆ ] ≤ C B κ N , t , M , κ N , t , M = N t M R 4 q R /2 . (81) For any deterministic weights 0 ≤ w h ≤ 1 0\le w_h\le1 0 ≤ w h ≤ 1
1 M ∑ h < M w h ∣ V h 2 − X h 2 ∣ ≤ 2 E N ⋆ Q N , t , M c y c + ( E N ⋆ ) 2 , (82) \frac{1}{M}\sum_{h<M}w_h|V_h^2-X_h^2|\le2E_N^\star\sqrt{Q^{\mathrm{cyc}}_{N,t,M}}+(E_N^\star)^2, \tag*{(82)} M 1 h < M ∑ w h ∣ V h 2 − X h 2 ∣ ≤ 2 E N ⋆ Q N , t , M cyc + ( E N ⋆ ) 2 , (82) sup 0 ≤ u ≤ 1 ∣ 1 M ∑ h < ⌊ M u ⌋ w h ( V h 2 − X h 2 ) ∣ ≤ M { 2 E N ⋆ Q N , t , M c y c + ( E N ⋆ ) 2 } . (83) \sup_{0\le u\le1}\left|\frac{1}{\sqrt{M}}\sum_{h<\lfloor Mu\rfloor}w_h(V_h^2-X_h^2)\right|\le\sqrt{M}\{2E_N^\star\sqrt{Q^{\mathrm{cyc}}_{N,t,M}}+(E_N^\star)^2\}. \tag*{(83)} 0 ≤ u ≤ 1 sup M 1 h < ⌊ M u ⌋ ∑ w h ( V h 2 − X h 2 ) ≤ M { 2 E N ⋆ Q N , t , M cyc + ( E N ⋆ ) 2 } . (83) The path bound also holds after interpolation. If t → ∞ t\to\infty t → ∞ τ N , R → 0 \tau_{N,R}\to0 τ N , R → 0
Proof. By (9.2), the error for explicit centring is L N ( t + h ) / t L_N(t+h)/\sqrt{t} L N ( t + h ) / t R N , k \mathcal{R}_{N,k} R N , k
E [ ∣ L N ( s ) ∣ ] ≤ C B N ∑ k = 3 s ∣ a s , k ∣ k 4 q k / 2 ≤ C B R 4 q R / 2 / N , s ≤ R . \mathbb{E}[|L_N(s)|]\le\frac{C_B}{N}\sum_{k=3}^{s}|a_{s,k}|k^4q^{k/2}\le C_BR^4q^{R/2}/N,\qquad s\le R. E [ ∣ L N ( s ) ∣ ] ≤ N C B k = 3 ∑ s ∣ a s , k ∣ k 4 q k /2 ≤ C B R 4 q R /2 / N , s ≤ R . For exact-mean centring use also (9.3). Bounding the maximum by the sum proves (9.5). The identity V h 2 − X h 2 = 2 X h ( V h − X h ) + ( V h − X h ) 2 V_h^2-X_h^2=2X_h(V_h-X_h)+(V_h-X_h)^2 V h 2 − X h 2 = 2 X h ( V h − X h ) + ( V h − X h ) 2 h h h
κ N , t , M τ N , R = M t q R / 2 ≤ C R q − R / 2 , M κ N , t , M τ N , R ≤ C R q − R / 2 . (84) \frac{\kappa_{N,t,M}}{\tau_{N,R}}=\frac{M}{\sqrt{t}q^{R/2}}\le C\sqrt{R}q^{-R/2},\qquad\frac{\sqrt{M}\kappa_{N,t,M}}{\tau_{N,R}}\le CRq^{-R/2}. \tag*{(84)} τ N , R κ N , t , M = t q R /2 M ≤ C R q − R /2 , τ N , R M κ N , t , M ≤ C R q − R /2 . (84) Thus M E N ⋆ → 0 \sqrt{M}E_N^\star\to0 M E N ⋆ → 0 Q N , t , M c y c Q^{\mathrm{cyc}}_{N,t,M} Q N , t , M cyc Q N , t , M c y c Q^{\mathrm{cyc}}_{N,t,M} Q N , t , M cyc ( M E N ⋆ ) 2 / M (\sqrt{M}E_N^\star)^2/\sqrt{M} ( M E N ⋆ ) 2 / M
The choices w h = 1 w_h=1 w h = 1 w h = t / ( t + h ) w_h=t/(t+h) w h = t / ( t + h )
For observation-time normalisation, keeping each observation time in the remainder estimate removes the restriction on the relative window length. Put
X ~ h = 1 t + h ∑ j = 3 R β t + h , j ( C N , j − λ j ) , Q ~ N , t , M = 1 M ∑ h < M X ~ h 2 . \widetilde{X}_h=\frac{1}{\sqrt{t+h}}\sum_{j=3}^{R}\beta_{t+h,j}(C_{N,j}-\lambda_j),\qquad\widetilde{Q}_{N,t,M}=\frac{1}{M}\sum_{h<M}\widetilde{X}_h^2. X h = t + h 1 j = 3 ∑ R β t + h , j ( C N , j − λ j ) , Q N , t , M = M 1 h < M ∑ X h 2 . Lemma 9.3 (Observation-time transfer on unrestricted windows). Let t , M ≥ 1 t,M\ge1 t , M ≥ 1 R = t + M ≥ 3 R=t+M\ge3 R = t + M ≥ 3 N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2
E ~ N ⋆ = max h < M ∣ A N ⋆ ( t + h ) t + h − X ~ h ∣ . \widetilde{E}^{\star}_N=\max_{h<M}\left|\frac{A_N^{\star}(t+h)}{\sqrt{t+h}}-\widetilde{X}_h\right|. E N ⋆ = h < M max t + h A N ⋆ ( t + h ) − X h . Then
E [ E ~ N ⋆ ] ≤ C B R 7 / 2 q R / 2 N , M E [ E ~ N ⋆ ] ≤ C B τ N , R q − R / 2 . (85) \mathbb{E}[\widetilde{E}^{\star}_N]\le C_B\frac{R^{7/2}q^{R/2}}{N},\qquad\sqrt{M}\mathbb{E}[\widetilde{E}^{\star}_N]\le C_B\tau_{N,R}q^{-R/2}. \tag*{(85)} E [ E N ⋆ ] ≤ C B N R 7/2 q R /2 , M E [ E N ⋆ ] ≤ C B τ N , R q − R /2 . (85) On each realisation, the difference between the squared observation-time trace process and its cycle counterpart is bounded in supremum norm by
M { 2 E ~ N ⋆ Q ~ N , t , M + ( E ~ N ⋆ ) 2 } . (86) \sqrt{M}\left\{2\widetilde{E}^{\star}_N\sqrt{\widetilde{Q}_{N,t,M}}+(\widetilde{E}^{\star}_N)^2\right\}. \tag*{(86)} M { 2 E N ⋆ Q N , t , M + ( E N ⋆ ) 2 } . (86) The bound also holds for the interpolations. If M → ∞ M\to\infty M → ∞ τ N , R → 0 \tau_{N,R}\to0 τ N , R → 0 M / t M/t M / t
Proof. Use the proof of Lemma 9.2 at each time s = t + h s=t+h s = t + h s s s R R R (79) , give
E [ ∣ A N ⋆ ( s ) s − 1 s ∑ j ≥ 3 β s , j ( C N , j − λ j ) ∣ ] ≤ C B s 7 / 2 q s / 2 N . \mathbb{E}\left[\left|\frac{A_N^{\star}(s)}{\sqrt{s}}-\frac{1}{\sqrt{s}}\sum_{j\ge3}\beta_{s,j}(C_{N,j}-\lambda_j)\right|\right]\le\frac{C_Bs^{7/2}q^{s/2}}{N}. E [ s A N ⋆ ( s ) − s 1 j ≥ 3 ∑ β s , j ( C N , j − λ j ) ] ≤ N C B s 7/2 q s /2 . For s = 1 , 2 s=1,2 s = 1 , 2
∑ s = t t + M − 1 s 7 / 2 q s / 2 ≤ C q R 7 / 2 q R / 2 . \sum_{s=t}^{t+M-1}s^{7/2}q^{s/2}\le C_qR^{7/2}q^{R/2}. s = t ∑ t + M − 1 s 7/2 q s /2 ≤ C q R 7/2 q R /2 . Since M ≤ R M\le R M ≤ R (85) . The difference-of-squares identity and Cauchy–Schwarz prove (86) , exactly as in (83) .
The ideal counterpart Q t , M o = M − 1 ∑ h < M P t + h 2 / ( t + h ) Q^o_{t,M}=M^{-1}\sum_{h<M}P_{t+h}^2/(t+h) Q t , M o = M − 1 ∑ h < M P t + h 2 / ( t + h ) Var ( P s ) ≤ C B s \operatorname{Var}(P_s)\le C_Bs Var ( P s ) ≤ C B s s ≥ 1 s\ge1 s ≥ 1 A > 0 A>0 A > 0
P ( Q ~ N , t , M > A ) ≤ P ( Q t , M o > A ) + C B τ N , R ≤ C B / A + C B τ N , R . \mathbb{P}(\widetilde{Q}_{N,t,M}>A)\le\mathbb{P}(Q^o_{t,M}>A)+C_B\tau_{N,R}\le C_B/A+C_B\tau_{N,R}. P ( Q N , t , M > A ) ≤ P ( Q t , M o > A ) + C B τ N , R ≤ C B / A + C B τ N , R . Thus Q ~ N , t , M \widetilde{Q}_{N,t,M} Q N , t , M (85) gives M E ~ N ⋆ → 0 \sqrt{M}\widetilde{E}^{\star}_N\to0 M E N ⋆ → 0 (86) tend to zero. Interpolation takes convex combinations of neighbouring grid values. ∎
Ideal moments and quantitative common-time averages
In this subsection, the common-time bounds require p B = 1 p_B = 1 p B = 1
e ( t , M ) = M t , Q t , M = 1 M t ∑ h < M P t + h 2 . e(t,M)=\frac{M}{t}, \qquad Q_{t,M}=\frac{1}{Mt}\sum_{h<M}P_{t+h}^{2}. e ( t , M ) = t M , Q t , M = M t 1 h < M ∑ P t + h 2 . For a non-bipartite base, Theorem 6.5 and Lemma 7.2 imply, with one fixed constant C B C_B C B
max a , b < M ∣ K t ( a , b ) − Γ d ( a − b ) ∣ ≤ D B ( t , M ) , D B ( t , M ) = C B { e ( t , M ) + t − 1 } , (87) \max_{a,b<M}\left|K_t(a,b)-\Gamma_d(a-b)\right|\le D_B(t,M), \qquad D_B(t,M)=C_B\{e(t,M)+t^{-1}\}, \tag*{(87)} a , b < M max ∣ K t ( a , b ) − Γ d ( a − b ) ∣ ≤ D B ( t , M ) , D B ( t , M ) = C B { e ( t , M ) + t − 1 } , (87) where K t ( a , b ) = t − 1 Cov ( P t + a , P t + b ) K_t(a,b)=t^{-1}\operatorname{Cov}(P_{t+a},P_{t+b}) K t ( a , b ) = t − 1 Cov ( P t + a , P t + b )
Proposition 9.4 (Ideal second and fourth moments). For every base and all nonnegative s , u s,u s , u
Cov ( P s 2 , P u 2 ) = 2 Cov ( P s , P u ) 2 + ρ B ( s , u ) , \operatorname{Cov}(P_s^2,P_u^2)=2\operatorname{Cov}(P_s,P_u)^2+\rho_B(s,u), Cov ( P s 2 , P u 2 ) = 2 Cov ( P s , P u ) 2 + ρ B ( s , u ) , ρ B ( s , u ) = ∑ j ≥ 3 λ j β s , j 2 β u , j 2 ∈ [ 0 , C B ] . (88) \rho_B(s,u)=\sum_{j\ge3}\lambda_j\beta_{s,j}^{2}\beta_{u,j}^{2}\in[0,C_B]. \tag*{(88)} ρ B ( s , u ) = j ≥ 3 ∑ λ j β s , j 2 β u , j 2 ∈ [ 0 , C B ] . (88) For p B = 1 p_B=1 p B = 1 t ≥ 4 t\ge4 t ≥ 4 1 ≤ M ≤ t / 2 1\le M\le t/2 1 ≤ M ≤ t /2
∣ E [ Q t , M ] − c d ∣ ≤ D B ( t , M ) , E [ ∣ Q t , M − c d ∣ 2 ] ≤ C B { M − 1 + t − 2 + e ( t , M ) 2 } . (89) \left|\mathbb{E}[Q_{t,M}]-c_d\right|\le D_B(t,M), \qquad\mathbb{E}\left[|Q_{t,M}-c_d|^2\right]\le C_B\{M^{-1}+t^{-2}+e(t,M)^2\}. \tag*{(89)} ∣ E [ Q t , M ] − c d ∣ ≤ D B ( t , M ) , E [ ∣ Q t , M − c d ∣ 2 ] ≤ C B { M − 1 + t − 2 + e ( t , M ) 2 } . (89) If also t , M → ∞ t,M\to\infty t , M → ∞
M { e ( t , M ) + t − 1 } ⟶ 0 , (90) \sqrt{M}\{e(t,M)+t^{-1}\}\longrightarrow0, \tag*{(90)} M { e ( t , M ) + t − 1 } ⟶ 0 , (90) then M Var ( Q t , M ) → 4 c d M\operatorname{Var}(Q_{t,M})\to4c_d M Var ( Q t , M ) → 4 c d M = ⌊ t γ ⌋ M=\lfloor t^\gamma\rfloor M = ⌊ t γ ⌋ 0 < γ < 2 / 3 0<\gamma<2/3 0 < γ < 2/3
Proof. The joint cumulant-generating function is ∑ j λ j { exp ( ∑ a z a β s a , j ) − 1 − ∑ a z a β s a , j } \sum_j\lambda_j\{\exp(\sum_a z_a\beta_{s_a,j})-1-\sum_a z_a\beta_{s_a,j}\} ∑ j λ j { exp ( ∑ a z a β s a , j ) − 1 − ∑ a z a β s a , j } B j B_j B j λ j ≤ m 0 q j / j \lambda_j\le m_0q^j/j λ j ≤ m 0 q j / j ∑ j λ j B j 4 ≤ C B ∑ j j 3 q − j < ∞ \sum_j\lambda_jB_j^4\le C_B\sum_jj^3q^{-j}<\infty ∑ j λ j B j 4 ≤ C B ∑ j j 3 q − j < ∞
Var ( Q t , M ) = 2 M 2 ∑ a , b < M K t ( a , b ) 2 + O B ( t − 2 ) . \operatorname{Var}(Q_{t,M})=\frac{2}{M^2}\sum_{a,b<M}K_t(a,b)^2+O_B(t^{-2}). Var ( Q t , M ) = M 2 2 a , b < M ∑ K t ( a , b ) 2 + O B ( t − 2 ) . By Lemma 7.4 , the row sums of squared entries of ( Γ d ( a − b ) ) (\Gamma_d(a-b)) ( Γ d ( a − b )) K t ( a , b ) 2 ≤ 2 Γ d ( a − b ) 2 + 2 D B ( t , M ) 2 K_t(a,b)^2\le2\Gamma_d(a-b)^2+2D_B(t,M)^2 K t ( a , b ) 2 ≤ 2 Γ d ( a − b ) 2 + 2 D B ( t , M ) 2
2 M ∑ a , b < M Γ d ( a − b ) 2 = 2 ∑ ∣ h ∣ < M ( 1 − ∣ h ∣ / M ) Γ d ( h ) 2 ⟶ 4 c d . \frac{2}{M}\sum_{a,b<M}\Gamma_d(a-b)^2=2\sum_{|h|<M}(1-|h|/M)\Gamma_d(h)^2\longrightarrow4c_d. M 2 a , b < M ∑ Γ d ( a − b ) 2 = 2 ∣ h ∣ < M ∑ ( 1 − ∣ h ∣/ M ) Γ d ( h ) 2 ⟶ 4 c d . Replacing Γ d \Gamma_d Γ d K t K_t K t C B { D B ( t , M ) log ( 2 + M ) + M D B ( t , M ) 2 } C_B\{D_B(t,M)\log(2+M)+MD_B(t,M)^2\} C B { D B ( t , M ) log ( 2 + M ) + M D B ( t , M ) 2 } ∣ Γ d ( h ) ∣ ≤ C d / ( 1 + ∣ h ∣ ) |\Gamma_d(h)|\le C_d/(1+|h|) ∣ Γ d ( h ) ∣ ≤ C d / ( 1 + ∣ h ∣ ) O B ( M / t 2 ) O_B(M/t^2) O B ( M / t 2 ) M = ⌊ t γ ⌋ M=\lfloor t^\gamma\rfloor M = ⌊ t γ ⌋ M { e ( t , M ) + t − 1 } → 0 \sqrt{M}\{e(t,M)+t^{-1}\}\to0 M { e ( t , M ) + t − 1 } → 0 0 < γ < 2 / 3 0<\gamma<2/3 0 < γ < 2/3 □ \square □
Proposition 9.5 (Finite-size probability and mean bounds). Suppose p B = 1 p_B=1 p B = 1 t ≥ 4 t\ge4 t ≥ 4 1 ≤ M ≤ t / 2 1\le M\le t/2 1 ≤ M ≤ t /2 R = t + M R=t+M R = t + M N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 0 < η ≤ 1 0<\eta\le1 0 < η ≤ 1
P ( ∣ F ^ N , t , M ⋆ − c d ∣ > η ) ≤ C B { ( 1 + η − 1 ) τ N , R + η − 2 ( M − 1 + t − 2 + e ( t , M ) 2 ) } . (91) \mathbb{P}\left(|\widehat{F}^{\star}_{N,t,M}-c_d|>\eta\right)\le C_B\left\{(1+\eta^{-1})\tau_{N,R}+\eta^{-2}(M^{-1}+t^{-2}+e(t,M)^2)\right\}. \tag*{(91)} P ( ∣ F N , t , M ⋆ − c d ∣ > η ) ≤ C B { ( 1 + η − 1 ) τ N , R + η − 2 ( M − 1 + t − 2 + e ( t , M ) 2 ) } . (91) Uniformly for 0 ≤ h < M 0\le h<M 0 ≤ h < M
∣ E [ A N ⋆ ( t + h ) 2 t ] − c d ∣ ≤ C B { e ( t , M ) + t − 1 + δ N , R + 1 { ⋆ = ∗ } R 8 q R / ( N 2 t ) } . (92) \left|\mathbb{E}\left[\frac{A_N^\star(t+h)^2}{t}\right]-c_d\right|\le C_B\{e(t,M)+t^{-1}+\delta_{N,R}+\mathbf{1}_{\{\star=\ast\}}R^8q^R/(N^2t)\}. \tag*{(92)} E [ t A N ⋆ ( t + h ) 2 ] − c d ≤ C B { e ( t , M ) + t − 1 + δ N , R + 1 { ⋆ = ∗ } R 8 q R / ( N 2 t )} . (92) The same bound holds for ∣ E [ F ^ N , t , M ⋆ ] − c d ∣ |\mathbb{E}[\widehat{F}^{\star}_{N,t,M}]-c_d| ∣ E [ F N , t , M ⋆ ] − c d ∣ C B δ N , R C_B\delta_{N,R} C B δ N , R
Proof. For common-time weights, (82) gives ∣ F ^ ⋆ − Q c y c ∣ ≤ 2 E N ⋆ Q c y c + ( E N ⋆ ) 2 |\widehat{F}^{\star}-Q^{\mathrm{cyc}}|\le2E_N^{\star}\sqrt{Q^{\mathrm{cyc}}}+(E_N^{\star})^2 ∣ F ⋆ − Q cyc ∣ ≤ 2 E N ⋆ Q cyc + ( E N ⋆ ) 2 c d < 2 c_d<2 c d < 2 ∣ Q c y c − c d ∣ ≤ η / 2 |Q^{\mathrm{cyc}}-c_d|\le\eta/2 ∣ Q cyc − c d ∣ ≤ η /2 E N ⋆ ≤ η / 8 E_N^{\star}\le\eta/8 E N ⋆ ≤ η /8 η / 2 \eta/2 η /2 5 / 2 / 4 + 1 / 64 < 1 / 2 \sqrt{5}/2/4+1/64<1/2 5 /2/4 + 1/64 < 1/2
P ( ∣ F ^ ⋆ − c d ∣ > η ) ≤ 8 E [ E N ⋆ ] η + C B τ N , R + P ( ∣ Q t , M − c d ∣ > η / 2 ) . \mathbb{P}(|\widehat{F}^{\star}-c_d|>\eta)\le\frac{8\mathbb{E}[E_N^{\star}]}{\eta}+C_B\tau_{N,R}+\mathbb{P}(|Q_{t,M}-c_d|>\eta/2). P ( ∣ F ⋆ − c d ∣ > η ) ≤ η 8 E [ E N ⋆ ] + C B τ N , R + P ( ∣ Q t , M − c d ∣ > η /2 ) . Markov’s inequality, (81) , (84) and (89) prove (91) . For the mean bound, the error in (72) with s = u = t + h s=u=t+h s = u = t + h t t t C B δ N , R C_B\delta_{N,R} C B δ N , R R / t ≤ 3 / 2 R/t\le3/2 R / t ≤ 3/2 (87) gives (92) for exact-mean centring. For explicit centring use
E [ A ^ N ⋆ ( s ) 2 ] = Var ( Tr U G N ) + ( E [ Tr U G N ] − m N ( s ) ) 2 \mathbb{E}[\widehat{A}_N^{\star}(s)^2]=\operatorname{Var}(\operatorname{Tr}U_{G_N})+\left(\mathbb{E}[\operatorname{Tr}U_{G_N}]-m_N(s)\right)^2 E [ A N ⋆ ( s ) 2 ] = Var ( Tr U G N ) + ( E [ Tr U G N ] − m N ( s ) ) 2 and (79) . The extra term is absorbed since R 3 / ( N t ) ≤ 3 / 16 R^3/(Nt)\le3/16 R 3 / ( N t ) ≤ 3/16
As t , M → ∞ t,M\to\infty t , M → ∞ M / t → 0 M/t\to0 M / t → 0 τ N , R → 0 \tau_{N,R}\to0 τ N , R → 0 δ N , R → 0 \delta_{N,R}\to0 δ N , R → 0 L 1 L^1 L 1
Functional limits and temporal averaging
We prove the observation-time limit using a time-decaying Gaussian coupling and an invariance principle for quadratic arrays with a product bound on the covariance error. These estimates apply to arbitrary growing windows of positive times. We then transfer the limit to the finite lift, identify the common-time growth profile, and prove the law of large numbers directly.
A simultaneous Gaussian coupling
Write ∥ V ∥ L p = ( E [ ∣ V ∣ p ] ) 1 / p \|V\|_{L^p}=(\mathbb{E}[|V|^p])^{1/p} ∥ V ∥ L p = ( E [ ∣ V ∣ p ] ) 1/ p 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ ψ ( x ) = e ∣ x ∣ − 1 \psi(x)=e^{|x|}-1 ψ ( x ) = e ∣ x ∣ − 1 ∥ V ∥ ψ = inf { a > 0 : E [ ψ ( V / a ) ] ≤ 1 } \|V\|_{\psi}=\inf\{a>0:\mathbb{E}[\psi(V/a)]\le1\} ∥ V ∥ ψ = inf { a > 0 : E [ ψ ( V / a )] ≤ 1 } W ψ W_{\psi} W ψ [33 ] gives
sup λ ≥ 0 W ψ ( Poisson ( λ ) , N ( λ , λ ) ) < ∞ . \sup_{\lambda\ge0}W_{\psi}(\operatorname{Poisson}(\lambda),\mathcal{N}(\lambda,\lambda))<\infty. λ ≥ 0 sup W ψ ( Poisson ( λ ) , N ( λ , λ )) < ∞. For each prescribed λ j \lambda_j λ j λ j = 0 \lambda_j=0 λ j = 0 Z j = 0 Z_j=0 Z j = 0 ( Z j , ξ j ) (Z_j,\xi_j) ( Z j , ξ j )
∥ Z j − λ j − λ j ξ j ∥ ψ ≤ C , ∥ Z j − λ j − λ j ξ j ∥ L p ≤ C p ( 1 ≤ p < ∞ ) . (93) \|Z_j-\lambda_j-\sqrt{\lambda_j}\xi_j\|_{\psi}\le C,\qquad
\|Z_j-\lambda_j-\sqrt{\lambda_j}\xi_j\|_{L^p}\le C_p\quad(1\le p<\infty). \tag*{(93)} ∥ Z j − λ j − λ j ξ j ∥ ψ ≤ C , ∥ Z j − λ j − λ j ξ j ∥ L p ≤ C p ( 1 ≤ p < ∞ ) . (93) The second bound follows from x p ≤ C p e x x^p\le C_p e^x x p ≤ C p e x x ≥ 0 x\ge0 x ≥ 0 p p p
Proposition 10.1 (Time-decaying ideal trace coupling). Under the coupling in (93) , let
P s G = ∑ j ≥ 3 β s , j λ j ξ j . P_s^G=\sum_{j\ge3}\beta_{s,j}\sqrt{\lambda_j}\xi_j. P s G = j ≥ 3 ∑ β s , j λ j ξ j . For every fixed 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞
∥ sup s ≥ 1 s ∣ P s − P s G ∣ ∥ L p ≤ C p , d , Cov ( P s G , P u G ) = Cov ( P s , P u ) . (94) \left\|\sup_{s\ge1}\sqrt{s}\,|P_s-P_s^G|\right\|_{L^p}\le C_{p,d},\qquad\operatorname{Cov}(P_s^G,P_u^G)=\operatorname{Cov}(P_s,P_u). \tag*{(94)} s ≥ 1 sup s ∣ P s − P s G ∣ L p ≤ C p , d , Cov ( P s G , P u G ) = Cov ( P s , P u ) . (94) For all positive integers t , M t,M t , M d h d_h d h
∥ sup 0 ≤ v ≤ 1 ∣ 1 M ∑ h < ⌊ M v ⌋ P t + h 2 − ( P t + h G ) 2 d h ∣ ∥ L p ≤ C p , B M ∑ h < M 1 d h . (95) \left\|\sup_{0\le v\le1}\left|\frac{1}{\sqrt{M}}\sum_{h<\lfloor Mv\rfloor}\frac{P_{t+h}^2-(P_{t+h}^G)^2}{d_h}\right|\right\|_{L^p}
\le\frac{C_{p,B}}{\sqrt{M}}\sum_{h<M}\frac{1}{d_h}. \tag*{(95)} 0 ≤ v ≤ 1 sup M 1 h < ⌊ M v ⌋ ∑ d h P t + h 2 − ( P t + h G ) 2 L p ≤ M C p , B h < M ∑ d h 1 . (95) The same bound holds after interpolation. In particular, with
H t , M = ∑ h = 0 M − 1 1 t + h ≤ 1 + log M , (96) H_{t,M}=\sum_{h=0}^{M-1}\frac{1}{t+h}\le1+\log M, \tag*{(96)} H t , M = h = 0 ∑ M − 1 t + h 1 ≤ 1 + log M , (96) the observation-time error is at most C p , B H t , M / M C_{p,B}H_{t,M}/\sqrt{M} C p , B H t , M / M t ≥ 1 t\ge1 t ≥ 1 C p , B M / t C_{p,B}\sqrt{M}/t C p , B M / t
Proof. Apply Lemma 6.2 with amplitude A ( x ) = cos ( k x ) A(x)=\cos(kx) A ( x ) = cos ( k x ) f s f_s f s ∥ A ∥ ∞ + ∥ A ′ ∥ L 1 ≤ 1 + 4 k \lVert A\rVert_\infty+\lVert A'\rVert_{L^1}\le1+4k ∥ A ∥ ∞ + ∥ A ′ ∥ L 1 ≤ 1 + 4 k
∣ a s , k ∣ ≤ C d ( 1 + k ) s − 1 / 2 , s , k ≥ 1. |a_{s,k}|\le C_d(1+k)s^{-1/2},\qquad s,k\ge1. ∣ a s , k ∣ ≤ C d ( 1 + k ) s − 1/2 , s , k ≥ 1. For k > s k>s k > s z j = q − j / 2 z_j=q^{-j/2} z j = q − j /2 (61) therefore implies
s ∣ β s , j ∣ ≤ D j : = 2 C d j { z j 1 − z j + j z j ( 1 − z j ) 2 } , ∑ j ≥ 3 D j < ∞ . (97) \sqrt{s}\,|\beta_{s,j}|\le D_j:=2C_dj\left\{\frac{z_j}{1-z_j}+\frac{jz_j}{(1-z_j)^2}\right\},\qquad\sum_{j\ge3}D_j<\infty. \tag*{(97)} s ∣ β s , j ∣ ≤ D j := 2 C d j { 1 − z j z j + ( 1 − z j ) 2 j z j } , j ≥ 3 ∑ D j < ∞. (97) Writing ζ j = Z j − λ j − λ j ξ j \zeta_j=Z_j-\lambda_j-\sqrt{\lambda_j}\xi_j ζ j = Z j − λ j − λ j ξ j
sup s ≥ 1 s ∣ P s − P s G ∣ ≤ ∑ j ≥ 3 D j ∣ ζ j ∣ . \sup_{s\ge1}\sqrt{s}\,|P_s-P_s^G|\le\sum_{j\ge3}D_j|\zeta_j|. s ≥ 1 sup s ∣ P s − P s G ∣ ≤ j ≥ 3 ∑ D j ∣ ζ j ∣. Minkowski’s inequality, (93) and monotone convergence bound the right-hand side in every finite L p L^p L p (94) equal ∑ j λ j β s , j β u , j \sum_j\lambda_j\beta_{s,j}\beta_{u,j} ∑ j λ j β s , j β u , j
By Lemma 7.2 , Gaussian moments satisfy ∥ P s G ∥ L 2 p ≤ C p , B s \lVert P_s^G\rVert_{L^{2p}}\le C_{p,B}\sqrt{s} ∥ P s G ∥ L 2 p ≤ C p , B s s ≥ 1 s\ge1 s ≥ 1 P s P_s P s
∥ P s 2 − ( P s G ) 2 ∥ L p ≤ ∥ P s − P s G ∥ L 2 p ∥ P s + P s G ∥ L 2 p ≤ C p , B . \lVert P_s^2-(P_s^G)^2\rVert_{L^p}\le\lVert P_s-P_s^G\rVert_{L^{2p}}\lVert P_s+P_s^G\rVert_{L^{2p}}\le C_{p,B}. ∥ P s 2 − ( P s G ) 2 ∥ L p ≤ ∥ P s − P s G ∥ L 2 p ∥ P s + P s G ∥ L 2 p ≤ C p , B . Bounding a partial sum by the full sum of absolute values and applying Minkowski proves (95) . Interpolation preserves the bound. Finally, t + h ≥ h + 1 t+h\ge h+1 t + h ≥ h + 1 (96) . □
The same time decay controls the removal of fixed cycle lengths. For fixed L L L P s ≤ L = ∑ 3 ≤ j ≤ L β s , j ( Z j − λ j ) P_s^{\le L}=\sum_{3\le j\le L}\beta_{s,j}(Z_j-\lambda_j) P s ≤ L = ∑ 3 ≤ j ≤ L β s , j ( Z j − λ j ) (97) gives
∥ sup s ≥ 1 s ∣ P s ≤ L ∣ ∥ L p ≤ C p , B , L . \left\lVert\sup_{s\ge1}\sqrt{s}\,|P_s^{\le L}|\right\rVert_{L^p}\le C_{p,B,L}. s ≥ 1 sup s ∣ P s ≤ L ∣ L p ≤ C p , B , L . In particular, Hölder’s inequality gives, uniformly for s ≥ 1 s\ge1 s ≥ 1
∥ P s 2 − ( P s − P s ≤ L ) 2 ∥ L p ≤ ∥ P s ≤ L ∥ L 2 p ( 2 ∥ P s ∥ L 2 p + ∥ P s ≤ L ∥ L 2 p ) ≤ C p , B , L . \lVert P_s^2-(P_s-P_s^{\le L})^2\rVert_{L^p}\le\lVert P_s^{\le L}\rVert_{L^{2p}}\left(2\lVert P_s\rVert_{L^{2p}}+\lVert P_s^{\le L}\rVert_{L^{2p}}\right)\le C_{p,B,L}. ∥ P s 2 − ( P s − P s ≤ L ) 2 ∥ L p ≤ ∥ P s ≤ L ∥ L 2 p ( 2 ∥ P s ∥ L 2 p + ∥ P s ≤ L ∥ L 2 p ) ≤ C p , B , L . The L p L^p L p C p , B , L H t , M / M C_{p,B,L}H_{t,M}/\sqrt{M} C p , B , L H t , M / M ( Z j ) j > L (Z_j)_{j>L} ( Z j ) j > L ( Y k ) k = 3 L (Y_k)_{k=3}^L ( Y k ) k = 3 L ( Z j ) 3 ≤ j ≤ L (Z_j)_{3\le j\le L} ( Z j ) 3 ≤ j ≤ L
An invariance principle for Gaussian quadratic forms
For a matrix A A A ∥ A ∥ o p = sup ∥ x ∥ 2 = 1 ∥ A x ∥ 2 \lVert A\rVert_{\mathrm{op}}=\sup_{\lVert x\rVert_2=1}\lVert Ax\rVert_2 ∥ A ∥ op = sup ∥ x ∥ 2 = 1 ∥ A x ∥ 2 ∥ A ∥ F = ( ∑ a , b ∣ A ( a , b ) ∣ 2 ) 1 / 2 \lVert A\rVert_F=\left(\sum_{a,b}|A(a,b)|^2\right)^{1/2} ∥ A ∥ F = ( ∑ a , b ∣ A ( a , b ) ∣ 2 ) 1/2 M − 1 M-1 M − 1 M M M (94) , which is not exactly stationary. The covariance matrices may be singular, and no uniform lower bound on their eigenvalues is needed. The row length may tend to infinity along any sequence. No coupling between different rows is assumed.
Theorem 10.2 (Gaussian arrays with a product covariance-error bound). Let γ : Z → R \gamma:\mathbb{Z}\to\mathbb{R} γ : Z → R T M = ( γ ( a − b ) ) 0 ≤ a , b < M T_M=(\gamma(a-b))_{0\leq a,b<M} T M = ( γ ( a − b ) ) 0 ≤ a , b < M
sup M ∥ T M ∥ o p < ∞ , g 2 : = ∑ h ∈ Z γ ( h ) 2 < ∞ . \sup_M\|T_M\|_{\mathrm{op}}<\infty,\qquad g_2:=\sum_{h\in\mathbb{Z}}\gamma(h)^2<\infty. M sup ∥ T M ∥ op < ∞ , g 2 := h ∈ Z ∑ γ ( h ) 2 < ∞. Let X M X_M X M K M K_M K M w M , a w_{M,a} w M , a C C C
∣ K M ( a , b ) − γ ( a − b ) ∣ ≤ C w M , a w M , b , |K_M(a,b)-\gamma(a-b)|\leq Cw_{M,a}w_{M,b}, ∣ K M ( a , b ) − γ ( a − b ) ∣ ≤ C w M , a w M , b , H M : = ∑ a < M w M , a 2 = o ( M ) , sup M ∑ a < M w M , a 4 < ∞ . (98) H_M:=\sum_{a<M}w_{M,a}^2=o(\sqrt{M}),\qquad\sup_M\sum_{a<M}w_{M,a}^4<\infty. \tag*{(98)} H M := a < M ∑ w M , a 2 = o ( M ) , M sup a < M ∑ w M , a 4 < ∞. (98) Then
S M ( v ) = 1 M ∑ h < ⌊ M v ⌋ { X M , h 2 − K M ( h , h ) } ⟹ 2 g 2 W ( v ) S_M(v)=\frac{1}{\sqrt{M}}\sum_{h<\lfloor Mv\rfloor}\{X_{M,h}^2-K_M(h,h)\}\Longrightarrow\sqrt{2g_2}\,W(v) S M ( v ) = M 1 h < ⌊ M v ⌋ ∑ { X M , h 2 − K M ( h , h )} ⟹ 2 g 2 W ( v ) in D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) J 1 J_1 J 1 C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) K M ( h , h ) K_M(h,h) K M ( h , h ) γ ( 0 ) \gamma(0) γ ( 0 )
Proof. Matrix estimates. Write E M = K M − T M E_M=K_M-T_M E M = K M − T M x , y x,y x , y
∣ x T E M y ∣ ≤ C ( ∑ a ∣ x a ∣ w M , a ) ( ∑ b ∣ y b ∣ w M , b ) ≤ C H M . |x^{\mathsf T}E_My|\leq C\left(\sum_a|x_a|w_{M,a}\right)\left(\sum_b|y_b|w_{M,b}\right)\leq CH_M. ∣ x T E M y ∣ ≤ C ( a ∑ ∣ x a ∣ w M , a ) ( b ∑ ∣ y b ∣ w M , b ) ≤ C H M . It follows that
∥ E M ∥ o p ≤ C H M , ∥ E M ∥ F ≤ C H M , ∥ K M ∥ o p M ⟶ 0. (99) \|E_M\|_{\mathrm{op}}\leq CH_M,\qquad\|E_M\|_F\leq CH_M,\qquad\frac{\|K_M\|_{\mathrm{op}}}{\sqrt{M}}\longrightarrow0. \tag*{(99)} ∥ E M ∥ op ≤ C H M , ∥ E M ∥ F ≤ C H M , M ∥ K M ∥ op ⟶ 0. (99) Since ∥ T M ∥ F 2 ≤ M g 2 \|T_M\|_F^2\leq Mg_2 ∥ T M ∥ F 2 ≤ M g 2
1 M ∑ a , b < M ∣ K M ( a , b ) 2 − T M ( a , b ) 2 ∣ ≤ C H M M + C H M 2 M ⟶ 0. \frac{1}{M}\sum_{a,b<M}|K_M(a,b)^2-T_M(a,b)^2|\leq C\frac{H_M}{\sqrt{M}}+C\frac{H_M^2}{M}\longrightarrow0. M 1 a , b < M ∑ ∣ K M ( a , b ) 2 − T M ( a , b ) 2 ∣ ≤ C M H M + C M H M 2 ⟶ 0. Here the squares are entrywise squares, not matrix products.
Covariance of partial sums. Wick’s formula gives
Cov ( S M ( v ) , S M ( w ) ) = 2 M ∑ a < ⌊ M v ⌋ ∑ b < ⌊ M w ⌋ K M ( a , b ) 2 . \operatorname{Cov}(S_M(v),S_M(w))=\frac{2}{M}\sum_{a<\lfloor Mv\rfloor}\sum_{b<\lfloor Mw\rfloor}K_M(a,b)^2. Cov ( S M ( v ) , S M ( w )) = M 2 a < ⌊ M v ⌋ ∑ b < ⌊ M w ⌋ ∑ K M ( a , b ) 2 . Replacing each entrywise square by T M ( a , b ) 2 T_M(a,b)^2 T M ( a , b ) 2 o ( 1 ) o(1) o ( 1 ) h h h a − b = h a-b=h a − b = h M M M min ( v , w ) \min(v,w) min ( v , w ) γ ( h ) 2 \gamma(h)^2 γ ( h ) 2
Cov ( S M ( v ) , S M ( w ) ) ⟶ 2 g 2 min ( v , w ) . (100) \operatorname{Cov}(S_M(v),S_M(w))\longrightarrow2g_2\min(v,w). \tag*{(100)} Cov ( S M ( v ) , S M ( w )) ⟶ 2 g 2 min ( v , w ) . (100) Finite-dimensional convergence. A fixed linear combination of partial sums has the form
M − 1 / 2 { X M T D M X M − Tr ( D M K M ) } = d ∑ r θ M , r ( ξ r 2 − 1 ) , M^{-1/2}\{X_M^{\mathsf T}D_MX_M-\operatorname{Tr}(D_MK_M)\}\overset{\mathrm{d}}{=}\sum_r\theta_{M,r}(\xi_r^2-1), M − 1/2 { X M T D M X M − Tr ( D M K M )} = d r ∑ θ M , r ( ξ r 2 − 1 ) , where D M D_M D M ξ r \xi_r ξ r θ M , r \theta_{M,r} θ M , r M − 1 / 2 K M 1 / 2 D M K M 1 / 2 M^{-1/2}K_M^{1/2}D_MK_M^{1/2} M − 1/2 K M 1/2 D M K M 1/2 K M K_M K M (99) gives max r ∣ θ M , r ∣ → 0 \max_r|\theta_{M,r}|\to0 max r ∣ θ M , r ∣ → 0 2 ∑ r θ M , r 2 2\sum_r\theta_{M,r}^2 2 ∑ r θ M , r 2 (100) . For fixed real z z z
∑ r { − i z θ M , r − 1 2 log ( 1 − 2 i z θ M , r ) } = − z 2 ∑ r θ M , r 2 + O z ( max r ∣ θ M , r ∣ ∑ r θ M , r 2 ) . \sum_r\left\{-iz\theta_{M,r}-\frac{1}{2}\log(1-2iz\theta_{M,r})\right\}=-z^2\sum_r\theta_{M,r}^2+O_z\left(\max_r|\theta_{M,r}|\sum_r\theta_{M,r}^2\right). r ∑ { − i z θ M , r − 2 1 log ( 1 − 2 i z θ M , r ) } = − z 2 r ∑ θ M , r 2 + O z ( r max ∣ θ M , r ∣ r ∑ θ M , r 2 ) . Each logarithm is taken continuously from value zero at z = 0 z=0 z = 0 L 2 L^2 L 2
Tightness of the interpolations. For a centred Gaussian quadratic form Q = ∑ r θ r ( ξ r 2 − 1 ) Q=\sum_r\theta_r(\xi_r^2-1) Q = ∑ r θ r ( ξ r 2 − 1 )
E [ Q 4 ] = 12 ( ∑ r θ r 2 ) 2 + 48 ∑ r θ r 4 ≤ 15 Var ( Q ) 2 . \mathbb{E}[Q^4]=12\left(\sum_r\theta_r^2\right)^2+48\sum_r\theta_r^4\leq15\operatorname{Var}(Q)^2. E [ Q 4 ] = 12 ( r ∑ θ r 2 ) 2 + 48 r ∑ θ r 4 ≤ 15 Var ( Q ) 2 . For 0 ≤ v < w ≤ 1 0\leq v<w\leq1 0 ≤ v < w ≤ 1
Q = 1 M ∑ a < M ω a { X M , a 2 − K M ( a , a ) } , 0 ≤ ω a ≤ 1 , ∑ a ω a = M ( w − v ) . Q=\frac{1}{\sqrt{M}}\sum_{a<M}\omega_a\{X^2_{M,a}-K_M(a,a)\},\qquad0\leq\omega_a\leq1,\qquad\sum_a\omega_a=M(w-v). Q = M 1 a < M ∑ ω a { X M , a 2 − K M ( a , a )} , 0 ≤ ω a ≤ 1 , a ∑ ω a = M ( w − v ) . The weight ω a \omega_a ω a M M M a a a K M ( a , b ) 2 ≤ 2 T M ( a , b ) 2 + 2 C 2 w M , a 2 w M , b 2 K_M(a,b)^2\leq2T_M(a,b)^2+2C^2w_{M,a}^2w_{M,b}^2 K M ( a , b ) 2 ≤ 2 T M ( a , b ) 2 + 2 C 2 w M , a 2 w M , b 2
Var ( Q ) ≤ C M { ∑ a ω a + ( ∑ a ω a w M , a 2 ) 2 } ≤ C M ∑ a ω a = C ( w − v ) . \begin{aligned}
\operatorname{Var}(Q)\leq\frac{C}{M}\left\{\sum_a\omega_a+\left(\sum_a\omega_aw_{M,a}^2\right)^2\right\} \\
&\leq\frac{C}{M}\sum_a\omega_a=C(w-v).
\end{aligned} Var ( Q ) ≤ M C ⎩ ⎨ ⎧ a ∑ ω a + ( a ∑ ω a w M , a 2 ) 2 ⎭ ⎬ ⎫ ≤ M C a ∑ ω a = C ( w − v ) . The first term uses ∑ b γ ( a − b ) 2 ≤ g 2 \sum_b\gamma(a-b)^2\leq g_2 ∑ b γ ( a − b ) 2 ≤ g 2 ω b ≤ 1 \omega_b\leq1 ω b ≤ 1 W 4 = sup M ∑ a < M w M , a 4 < ∞ W_4=\sup_M\sum_{a<M}w_{M,a}^4<\infty W 4 = sup M ∑ a < M w M , a 4 < ∞
( ∑ a ω a w M , a 2 ) 2 ≤ ( ∑ a ω a ) ( ∑ a ω a w M , a 4 ) ≤ W 4 ∑ a ω a . \left(\sum_a\omega_aw_{M,a}^2\right)^2\leq\left(\sum_a\omega_a\right)\left(\sum_a\omega_aw_{M,a}^4\right)\leq W_4\sum_a\omega_a. ( a ∑ ω a w M , a 2 ) 2 ≤ ( a ∑ ω a ) ( a ∑ ω a w M , a 4 ) ≤ W 4 a ∑ ω a . More explicitly, Var ( Q ) ≤ 4 ( g 2 + C 2 W 4 ) ( w − v ) \operatorname{Var}(Q)\leq4(g_2+C^2W_4)(w-v) Var ( Q ) ≤ 4 ( g 2 + C 2 W 4 ) ( w − v ) C C C H M = o ( M ) H_M=o(\sqrt{M}) H M = o ( M )
E [ ∣ S ‾ M ( w ) − S ‾ M ( v ) ∣ 4 ] ≤ C ∣ w − v ∣ 2 . \mathbb{E}[|\overline{S}_M(w)-\overline{S}_M(v)|^4]\leq C|w-v|^2. E [ ∣ S M ( w ) − S M ( v ) ∣ 4 ] ≤ C ∣ w − v ∣ 2 . This holds also when either endpoint lies between grid times, so the continuous-process tightness criterion applies.
Step processes and diagonal centring. The fourth-power weight bound implies sup M , a w M , a 2 < ∞ \sup_{M,a}w_{M,a}^2<\infty sup M , a w M , a 2 < ∞ sup M , a K M ( a , a ) < ∞ \sup_{M,a}K_M(a,a)<\infty sup M , a K M ( a , a ) < ∞ O ( M − 2 ) O(M^{-2}) O ( M − 2 )
P ( max a < M ∣ X M , a 2 − K M ( a , a ) ∣ M > η ) ≤ C M η 4 ⟶ 0. \mathbb{P}\left(\max_{a<M}\frac{|X^2_{M,a}-K_M(a,a)|}{\sqrt{M}}>\eta\right)\leq\frac{C}{M\eta^4}\longrightarrow0. P ( a < M max M ∣ X M , a 2 − K M ( a , a ) ∣ > η ) ≤ M η 4 C ⟶ 0. Thus the step process and its interpolation have the same continuous limit. Finally, replacing K M ( a , a ) K_M(a,a) K M ( a , a ) γ ( 0 ) \gamma(0) γ ( 0 ) C H M / M → 0 CH_M/\sqrt{M}\to0 C H M / M → 0
The uniform-error assumption max a , b ∣ K M ( a , b ) − γ ( a − b ) ∣ ≤ ϵ M \max_{a,b}|K_M(a,b)-\gamma(a-b)|\leq\epsilon_M max a , b ∣ K M ( a , b ) − γ ( a − b ) ∣ ≤ ϵ M M ϵ M → 0 \sqrt{M}\epsilon_M\to0 M ϵ M → 0 w M , a = ϵ M w_{M,a}=\sqrt{\epsilon_M} w M , a = ϵ M
The ideal observation-time limit
Define S t , M P , b ( u ) = M − 1 / 2 ∑ h < ⌊ M u ⌋ ( P t + h 2 / ( t + h ) − c d ) S^{P,b}_{t,M}(u)=M^{-1/2}\sum_{h<\lfloor Mu\rfloor}(P^2_{t+h}/(t+h)-c_d) S t , M P , b ( u ) = M − 1/2 ∑ h < ⌊ M u ⌋ ( P t + h 2 / ( t + h ) − c d )
Proposition 10.3 (Ideal limit for either period). For any sequence of positive integer pairs ( t , M ) (t,M) ( t , M ) M → ∞ M\to\infty M → ∞ S t , M P , b ⟹ 2 β c d W S^{P,b}_{t,M}\Longrightarrow2\sqrt{\beta c_d}\,W S t , M P , b ⟹ 2 β c d W D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) J 1 J_1 J 1 C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) ( Y k ) k = 3 ℓ (Y_k)_{k=3}^{\ell} ( Y k ) k = 3 ℓ
Proof. Proposition 10.1, with d h = t + h d_h=t+h d h = t + h L p L^p L p c p , B H t , M / M c_{p,B}H_{t,M}/\sqrt{M} c p , B H t , M / M t ≥ 1 t\ge1 t ≥ 1
Suppose first that B B B X M , a = P t + a G / t + a X_{M,a}=P^{\mathrm{G}}_{t+a}/\sqrt{t+a} X M , a = P t + a G / t + a (69) gives
∣ K M ( a , b ) − Γ d ( a − b ) ∣ ≤ C B ( t + a ) ( t + b ) . \left|K_M(a,b)-\Gamma_d(a-b)\right|\le\frac{C_B}{\sqrt{(t+a)(t+b)}}. ∣ K M ( a , b ) − Γ d ( a − b ) ∣ ≤ ( t + a ) ( t + b ) C B . Take w M , a = ( t + a ) − 1 / 2 w_{M,a}=(t+a)^{-1/2} w M , a = ( t + a ) − 1/2
∑ a < M w M , a 2 = H t , M ≤ 1 + log M = o ( M ) , ∑ a < M w M , a 4 ≤ ∑ n ≥ 1 n − 2 < ∞ . \sum_{a<M}w_{M,a}^{2}=H_{t,M}\le1+\log M=o(\sqrt{M}),\qquad\sum_{a<M}w_{M,a}^{4}\le\sum_{n\ge1}n^{-2}<\infty. a < M ∑ w M , a 2 = H t , M ≤ 1 + log M = o ( M ) , a < M ∑ w M , a 4 ≤ n ≥ 1 ∑ n − 2 < ∞. Theorem 10.2 and Lemma 7.4 therefore give the Brownian limit with variance rate 4 c d 4c_d 4 c d
For a bipartite base, let r t = 0 r_t=0 r t = 0 t t t r t = 1 r_t=1 r t = 1
s j = t + r t + 2 j , 0 ≤ j < m , m = # { 0 ≤ h < M : t + h is even } . s_j=t+r_t+2j,\qquad0\le j<m,\qquad m=\#\{0\le h<M:t+h\text{ is even}\}. s j = t + r t + 2 j , 0 ≤ j < m , m = # { 0 ≤ h < M : t + h is even } . Here m / M → 1 / 2 m/M\to1/2 m / M → 1/2 t t t M M M C B / s j s k C_B/\sqrt{s_js_k} C B / s j s k
γ 2 ( j − k ) = 2 Γ d ( 2 ( j − k ) ) . \gamma_2(j-k)=2\Gamma_d(2(j-k)). γ 2 ( j − k ) = 2 Γ d ( 2 ( j − k )) . The corresponding Toeplitz matrices have operator norm at most 4 4 4 2 2 2
∑ j ∈ Z γ 2 ( j ) 2 = 4 ∑ j ∈ Z Γ d ( 2 j ) 2 = 8 c d . \sum_{j\in\mathbb{Z}}\gamma_2(j)^2=4\sum_{j\in\mathbb{Z}}\Gamma_d(2j)^2=8c_d. j ∈ Z ∑ γ 2 ( j ) 2 = 4 j ∈ Z ∑ Γ d ( 2 j ) 2 = 8 c d . With w m , j = s j − 1 / 2 w_{m,j}=s_j^{-1/2} w m , j = s j − 1/2 1 + log ( 2 m ) 1+\log(2m) 1 + log ( 2 m ) ∑ n ≥ 1 n − 2 \sum_{n\ge1}n^{-2} ∑ n ≥ 1 n − 2
V m ( v ) = 1 m ∑ j < ⌊ m v ⌋ { ( P s j G ) 2 / s j − 2 c d } ⟹ 4 c d W ( v ) . V_m(v)=\frac{1}{\sqrt{m}}\sum_{j<\lfloor mv\rfloor}\left\{(P^{\mathrm{G}}_{s_j})^2/s_j-2c_d\right\}\Longrightarrow4\sqrt{c_d}\,W(v). V m ( v ) = m 1 j < ⌊ m v ⌋ ∑ { ( P s j G ) 2 / s j − 2 c d } ⟹ 4 c d W ( v ) . For n = ⌊ M u ⌋ n=\lfloor Mu\rfloor n = ⌊ M u ⌋ j ( n ) j(n) j ( n ) n n n
m / M V m ( j ( n ) / m ) + c d M { 2 j ( n ) − n } . (101) \sqrt{m/M}\,V_m(j(n)/m)+\frac{c_d}{\sqrt{M}}\{2j(n)-n\}. \tag*{(101)} m / M V m ( j ( n ) / m ) + M c d { 2 j ( n ) − n } . (101) The second term is bounded uniformly by c d / M c_d/\sqrt{M} c d / M ∣ 2 j ( n ) − n ∣ ≤ 1 |2j(n)-n|\le1 ∣2 j ( n ) − n ∣ ≤ 1 ∣ 2 m − M ∣ ≤ 1 |2m-M|\le1 ∣2 m − M ∣ ≤ 1 m ≥ 1 m\ge1 m ≥ 1
sup u ∈ [ 0 , 1 ] ∣ j ( ⌊ M u ⌋ ) / m − u ∣ ≤ m − 1 + M − 1 ≤ C / M . \sup_{u\in[0,1]}\left|j(\lfloor Mu\rfloor)/m-u\right|\le m^{-1}+M^{-1}\le C/M. u ∈ [ 0 , 1 ] sup ∣ j (⌊ M u ⌋) / m − u ∣ ≤ m − 1 + M − 1 ≤ C / M . Since M → ∞ M\to\infty M → ∞ V m V_m V m m / M \sqrt{m/M} m / M 1 / 2 1/\sqrt{2} 1/ 2 2 2 c d W 2\sqrt{2c_d}\,W 2 2 c d W c d / M c_d/\sqrt{M} c d / M
(95) transfers both cases to the Poisson field. Delete the finitely many cycle terms with j ≤ L j\le L j ≤ L C B , L H t , M / M C_{B,L}H_{t,M}/\sqrt{M} C B , L H t , M / M L 1 L^1 L 1 ( Y k ) k = 3 ∞ (Y_k)_{k=3}^\infty ( Y k ) k = 3 ∞
Transfer to the finite lift
Proof of Theorem 1.1. Write t = t N t=t_N t = t N M = M N M=M_N M = M N R = R N R=R_N R = R N R → ∞ R\to\infty R → ∞ N ≥ 8 R 2 N\ge8R^2 N ≥ 8 R 2 R ≥ L R\ge L R ≥ L
S N , t , M c y c , ♮ ( u ) = 1 M ∑ h < ⌊ M u ⌋ { X ~ h 2 − c d } . S^{\mathrm{cyc},\natural}_{N,t,M}(u)=\frac{1}{\sqrt{M}}\sum_{h<\lfloor Mu\rfloor}\{\widetilde{X}_h^2-c_d\}. S N , t , M cyc , ♮ ( u ) = M 1 h < ⌊ M u ⌋ ∑ { X h 2 − c d } . The complete-cycle-vector comparison shows that its law differs from that of S t , M P , ♮ S^{P,\natural}_{t,M} S t , M P , ♮ C B τ N , R C B^\tau N,R C B τ N , R Ψ k \Psi_k Ψ k (80) , whose fixed-degree proof needs no comparison of t t t M M M u = 1 u=1 u = 1 □ \square □
The common-time growth profile
Proof of Corollary 1.3. Suppress the subscripts and let U j = Z ♮ ( j / M ) U_j=Z^\natural(j/M) U j = Z ♮ ( j / M ) n = ⌊ M u ⌋ n=\lfloor Mu\rfloor n = ⌊ M u ⌋
∑ h = 0 n − 1 ( 1 + h / t ) ( U h + 1 − U h ) = ( 1 + n − 1 t ) U n − 1 t ∑ j = 1 n − 1 U j . (102) \sum_{h=0}^{n-1}(1+h/t)(U_{h+1}-U_h)=\left(1+\frac{n-1}{t}\right)U_n-\frac{1}{t}\sum_{j=1}^{n-1}U_j. \tag*{(102)} h = 0 ∑ n − 1 ( 1 + h / t ) ( U h + 1 − U h ) = ( 1 + t n − 1 ) U n − t 1 j = 1 ∑ n − 1 U j . (102) Both sides are zero when n = 0 n=0 n = 0 M / t → ρ < ∞ M/t\to\rho<\infty M / t → ρ < ∞ M → ∞ M\to\infty M → ∞ t → ∞ t\to\infty t → ∞ J 1 J_1 J 1 ρ σ B ∫ 0 u W ( v ) d v \rho\sigma_B\int_0^u W(v)\,dv ρ σ B ∫ 0 u W ( v ) d v ( 1 + ρ u ) σ B W ( u ) (1+\rho u)\sigma_B W(u) ( 1 + ρ u ) σ B W ( u )
( 1 + ρ u ) W ( u ) − ρ ∫ 0 u W ( v ) d v = ∫ 0 u ( 1 + ρ v ) d W ( v ) . (1+\rho u)W(u)-\rho\int_0^u W(v)\,dv=\int_0^u(1+\rho v)\,dW(v). ( 1 + ρ u ) W ( u ) − ρ ∫ 0 u W ( v ) d v = ∫ 0 u ( 1 + ρ v ) d W ( v ) . This proves the step-process limit jointly with the natural process. The profile-process jumps are the natural-process jumps multiplied by weights bounded by 1 + M / t 1+M/t 1 + M / t
The limiting stochastic integral is a centred continuous Gaussian process with independent increments and variance ∫ 0 u ( 1 + ρ v ) 2 d v = u + ρ u 2 + ρ 2 u 3 / 3 \int_0^u(1+\rho v)^2\,dv=u+\rho u^2+\rho^2u^3/3 ∫ 0 u ( 1 + ρ v ) 2 d v = u + ρ u 2 + ρ 2 u 3 /3 u = 1 u=1 u = 1 c d ( 1 + ( M − 1 ) / ( 2 t ) ) c_d(1+(M-1)/(2t)) c d ( 1 + ( M − 1 ) / ( 2 t )) □ \square □
The drift under common-time normalisation
Proof of Corollary 1.4. Fix one centring and suppress N , t , M , ⋆ N,t,M,\star N , t , M , ⋆ U j = Z ♮ ( j / M ) U_j=Z^\natural(j/M) U j = Z ♮ ( j / M ) n = ⌊ M u ⌋ n=\lfloor Mu\rfloor n = ⌊ M u ⌋
Z ( u ) − Z ♮ ( u ) = c d n ( n − 1 ) 2 t M + 1 t { ( n − 1 ) U n − ∑ j = 1 n − 1 U j } . (103) Z(u)-Z^\natural(u)=\frac{c_d n(n-1)}{2t\sqrt{M}}+\frac{1}{t}\left\{(n-1)U_n-\sum_{j=1}^{n-1}U_j\right\}. \tag*{(103)} Z ( u ) − Z ♮ ( u ) = 2 t M c d n ( n − 1 ) + t 1 { ( n − 1 ) U n − j = 1 ∑ n − 1 U j } . (103) For n = 0 n=0 n = 0 M − 1 / 2 ∑ h < n ( h / t ) A N ⋆ ( t + h 2 ) / ( t + h ) M^{-1/2}\sum_{h<n}(h/t)A_N^\star(t+h^2)/(t+h) M − 1/2 ∑ h < n ( h / t ) A N ⋆ ( t + h 2 ) / ( t + h ) M \sqrt{M} M U h + 1 − U h U_{h+1}-U_h U h + 1 − U h
The expression in braces in (103) is bounded in absolute value by 2 M ∥ Z ♮ ∥ ∞ 2M\lVert Z^\natural\rVert_\infty 2 M ∥ Z ♮ ∥ ∞ Theorem 1.1 , and M / t → 0 M/t \to0 M / t → 0 u u u
∣ n ( n − 1 ) 2 t M − M 3 / 2 2 t u 2 ∣ ≤ C M / t → 0. \left|\frac{n(n-1)}{2t\sqrt{M}}-\frac{M^{3/2}}{2t}u^2\right|\le C\sqrt{M}/t\to0. 2 t M n ( n − 1 ) − 2 t M 3/2 u 2 ≤ C M / t → 0. These two estimates prove (5) . Polygonal interpolation preserves the random-error bound; the uniform error in interpolating the quadratic drift is at most C / ( t M ) C/(t\sqrt{M}) C / ( t M ) M 3 / 2 / t → κ < ∞ M^{3/2}/t\to\kappa<\infty M 3/2 / t → κ < ∞ (103) at u = 1 u=1 u = 1 Z ♮ ( 1 ) Z^\natural(1) Z ♮ ( 1 ) Z ( 1 ) → + ∞ Z(1)\to+\infty Z ( 1 ) → + ∞
Self-averaging and averaged moment errors
The law of large numbers follows from direct ideal second moments and local-in-time finite-lift moment bounds. In particular, the expectation argument does not transfer unbounded moments through total variation.
Proof of Theorem 1.2. Write t = t N t=t_N t = t N M = M N M=M_N M = M N R = t + M R=t+M R = t + M H = H t , M H=H_{t,M} H = H t , M Q t , M ♯ = M − 1 ∑ h < M P t + h 2 / ( t + h ) Q^\sharp_{t,M}=M^{-1}\sum_{h<M}P^2_{t+h}/(t+h) Q t , M ♯ = M − 1 ∑ h < M P t + h 2 / ( t + h ) (68) gives
∣ E [ Q t , M ♯ ] − c d ∣ ≤ C B ( H + 1 ) / M . \left|\mathbb{E}[Q^\sharp_{t,M}]-c_d\right|\le C_B(H+1)/M. E [ Q t , M ♯ ] − c d ≤ C B ( H + 1 ) / M . Here the average of η B ( t + h ) \eta_B(t+h) η B ( t + h ) 1 / M 1/M 1/ M J B ( t + a , t + b ) J_B(t+a,t+b) J B ( t + a , t + b ) (69) , and the fourth-cumulant identity (88) , therefore imply
E [ ∣ Q t , M ♯ − c d ∣ 2 ] ≤ C B { 1 M + ( H + 1 ) 2 M 2 } → 0. \mathbb{E}\left[|Q^\sharp_{t,M}-c_d|^2\right]\le C_B\left\{\frac{1}{M}+\frac{(H+1)^2}{M^2}\right\}\to0. E [ ∣ Q t , M ♯ − c d ∣ 2 ] ≤ C B { M 1 + M 2 ( H + 1 ) 2 } → 0. To see this directly, write K a b = Cov ( P t + a , P t + b ) / ( t + a ) ( t + b ) K_{ab}=\operatorname{Cov}(P_{t+a},P_{t+b})/\sqrt{(t+a)(t+b)} K ab = Cov ( P t + a , P t + b ) / ( t + a ) ( t + b ) J a b = J B ( t + a , t + b ) J_{ab}=J_B(t+a,t+b) J ab = J B ( t + a , t + b )
Var ( Q t , M ♯ ) = 2 M 2 ∑ a , b < M K a b 2 + 1 M 2 ∑ a , b < M ρ B ( t + a , t + b ) ( t + a ) ( t + b ) . \operatorname{Var}(Q^\sharp_{t,M})=\frac{2}{M^2}\sum_{a,b<M}K_{ab}^2+\frac{1}{M^2}\sum_{a,b<M}\frac{\rho_B(t+a,t+b)}{(t+a)(t+b)}. Var ( Q t , M ♯ ) = M 2 2 a , b < M ∑ K ab 2 + M 2 1 a , b < M ∑ ( t + a ) ( t + b ) ρ B ( t + a , t + b ) . The row sums ∑ b J a b 2 \sum_bJ_{ab}^2 ∑ b J ab 2 ∣ K a b − J a b ∣ ≤ C B / ( t + a ) ( t + b ) |K_{ab}-J_{ab}|\le C_B/\sqrt{(t+a)(t+b)} ∣ K ab − J ab ∣ ≤ C B / ( t + a ) ( t + b ) K a b 2 ≤ 2 J a b 2 + 2 ( K a b − J a b ) 2 K_{ab}^2\le2J_{ab}^2+2(K_{ab}-J_{ab})^2 K ab 2 ≤ 2 J ab 2 + 2 ( K ab − J ab ) 2 ∑ a , b < M K a b 2 ≤ C B ( M + H 2 ) \sum_{a,b<M}K_{ab}^2\le C_B(M+H^2) ∑ a , b < M K ab 2 ≤ C B ( M + H 2 ) C B H 2 / M 2 C_BH^2/M^2 C B H 2 / M 2 (86) by M \sqrt{M} M F N , t , M ♯ F^\sharp_{N,t,M} F N , t , M ♯
For expectations, use (72) with s = u s=u s = u s s s s ≥ 3 s\ge3 s ≥ 3 s s s C B s 5 q s / N C_Bs^5q^s/N C B s 5 q s / N s = 1 , 2 s=1,2 s = 1 , 2
1 M ∑ s = t t + M − 1 ∣ Var ( A N 0 ( s ) ) − Var ( P s ) ∣ s ≤ C B N M ∑ s = t t + M − 1 s 5 q s ≤ C B δ N , R M . \frac{1}{M}\sum_{s=t}^{t+M-1}\frac{|\operatorname{Var}(A_N^0(s))-\operatorname{Var}(P_s)|}{s}\le\frac{C_B}{NM}\sum_{s=t}^{t+M-1}s^5q^s\le C_B\frac{\delta_{N,R}}{M}. M 1 s = t ∑ t + M − 1 s ∣ Var ( A N 0 ( s )) − Var ( P s ) ∣ ≤ N M C B s = t ∑ t + M − 1 s 5 q s ≤ C B M δ N , R . For explicit centring, (79) with cutoff s s s
C B N 2 M ∑ s = t t + M − 1 s 7 q s ≤ C B R 7 q R N 2 M = C B δ N , R R 2 M N . \frac{C_B}{N^2M}\sum_{s=t}^{t+M-1}s^7q^s\le C_B\frac{R^7q^R}{N^2M}=C_B\frac{\delta_{N,R}R^2}{M N}. N 2 M C B s = t ∑ t + M − 1 s 7 q s ≤ C B N 2 M R 7 q R = C B M N δ N , R R 2 . Consequently
∣ E [ F ^ N , t , M ♮ , ⋆ ] − c d ∣ ≤ C B { H + 1 M + δ N , R M + 1 { ⋆ = ∗ } R q t q N 2 M } . \left|\mathbb{E}\left[\widehat{F}^{\natural,\star}_{N,t,M}\right]-c_d\right|
\le C_B\left\{\frac{H+1}{M}+\frac{\delta_{N,R}}{M}+\mathbf{1}_{\{\star=\ast\}}\frac{R^q t^q}{N^2M}\right\}. E [ F N , t , M ♮ , ⋆ ] − c d ≤ C B { M H + 1 + M δ N , R + 1 { ⋆ = ∗ } N 2 M R q t q } . The cutoff assumption implies R 2 / N → 0 R^2/N\to0 R 2 / N → 0 δ N , R / M \delta_{N,R}/M δ N , R / M R 2 / N R^2/N R 2 / N F N F_N F N c ≥ 0 c\ge0 c ≥ 0
E [ ∣ F N − c ∣ ] = E [ F N ] + c − 2 E [ min ( F N , c ) ] . \mathbb{E}[|F_N-c|]=\mathbb{E}[F_N]+c-2\mathbb{E}[\min(F_N,c)]. E [ ∣ F N − c ∣ ] = E [ F N ] + c − 2 E [ min ( F N , c )] . Since min ( F N , c ) \min(F_N,c) min ( F N , c ) c c c E [ min ( F N , c ) ] → c \mathbb{E}[\min(F_N,c)]\to c E [ min ( F N , c )] → c L 1 L^1 L 1
For the common-time statistic assume M / t → ρ < ∞ M/t\to\rho<\infty M / t → ρ < ∞ v h = 1 + h / t v_h=1+h/t v h = 1 + h / t a t , M = M − 1 ∑ h < M v h = 1 + ( M − 1 ) / ( 2 t ) a_{t,M}=M^{-1}\sum_{h<M}v_h=1+(M-1)/(2t) a t , M = M − 1 ∑ h < M v h = 1 + ( M − 1 ) / ( 2 t )
Var ( 1 M ∑ h < M v h P t + h 2 t + h ) ≤ C B ( 1 + M / t ) 2 { 1 M + H 2 M 2 } ⟶ 0. \operatorname{Var}\left(\frac{1}{M}\sum_{h<M}v_h\frac{P^2_{t+h}}{t+h}\right)
\le C_B(1+M/t)^2\left\{\frac{1}{M}+\frac{H^2}{M^2}\right\}\longrightarrow0. Var ( M 1 h < M ∑ v h t + h P t + h 2 ) ≤ C B ( 1 + M / t ) 2 { M 1 + M 2 H 2 } ⟶ 0. Here 1 + M / t 1+M/t 1 + M / t
∣ 1 M ∑ h < M E [ P t + h 2 ] t − c d a t , M ∣ ≤ C B ( t − 1 + M − 1 ) . \left|\frac{1}{M}\sum_{h<M}\frac{\mathbb{E}[P^2_{t+h}]}{t}-c_da_{t,M}\right|
\le C_B(t^{-1}+M^{-1}). M 1 h < M ∑ t E [ P t + h 2 ] − c d a t , M ≤ C B ( t − 1 + M − 1 ) . For a bipartite base, the additional main term is an alternating sum of v h v_h v h C ( 1 + M / t ) C(1+M/t) C ( 1 + M / t ) M M M C B / t C_B/t C B / t c d ( 1 + ρ / 2 ) c_d(1+\rho/2) c d ( 1 + ρ /2 ) max h v h \max_h v_h max h v h max h v h \max_h v_h max h v h L 1 L^1 L 1
Fixed-offset fluctuations for bipartite bases
Time averages over consecutive observations have the same limit regardless of the initial-time parity. For fixed offsets, the odd-time coordinates are identically zero, and the initial-time parity determines the limiting vector.
Theorem 10.4 (Fixed-offset limit for a bipartite base). Suppose B B B r ≥ 1 r\ge1 r ≥ 1 h 1 , … , h r h_1,\ldots,h_r h 1 , … , h r H = max a ∣ h a ∣ H=\max_a|h_a| H = max a ∣ h a ∣ t N → ∞ t_N\to\infty t N → ∞ ε ∈ { 0 , 1 } \varepsilon\in\{0,1\} ε ∈ { 0 , 1 } R N = t N + H R_N=t_N+H R N = t N + H R N 4 q R N / N → 0 R_N^4q^{R_N}/N\to0 R N 4 q R N / N → 0
( A N ⋆ ( t N + h a ) / t N ) a = 1 r ⟹ ( 2 1 { ε + h a e v e n } G h a ) a = 1 r . \left(A_N^\star(t_N+h_a)/\sqrt{t_N}\right)_{a=1}^r
\Longrightarrow
\left(\sqrt{2}\mathbf{1}_{\{\varepsilon+h_a\ {\rm even}\}}G_{h_a}\right)_{a=1}^r. ( A N ⋆ ( t N + h a ) / t N ) a = 1 r ⟹ ( 2 1 { ε + h a even } G h a ) a = 1 r . where G G G R N 5 q R N / N → 0 R_N^5q^{R_N}/N\to0 R N 5 q R N / N → 0 W 2 W_2 W 2 t N q t N / N → 0 t_Nq^{t_N}/N\to0 t N q t N / N → 0 F N c ( t N ) / t N → 2 c d F_N^c(t_N)/t_N\to2c_d F N c ( t N ) / t N → 2 c d
Proof. The odd-time coordinates vanish by Lemma 7.1. For the even coordinates, apply (7.7) at reference time min a ( t N + h a ) \min_a(t_N+h_a) min a ( t N + h a ) t N t_N t N 2 Γ d ( h a − h b ) 2\Gamma_d(h_a-h_b) 2 Γ d ( h a − h b ) O ( t N − 1 / 2 ) O(t_N^{-1/2}) O ( t N − 1/2 ) Z j Z_j Z j
Proposition 8.1 transfers the joint law; its proof uses no primitivity. The deterministic difference between the two trace centrings is negligible by (79) and its first-moment proof. Under the additional condition, Proposition 8.2 and (48) give convergence of second moments. Weak convergence with convergent second moments yields the claimed W 2 W_2 W 2 Theorem 1.5 . The even-time form-factor conclusion is the diagonal case.
Common-time and expectation-centred consequences
Corollary 10.5 (Common-time consequences). Suppose p B = 1 \mathsf{p}_B=1 p B = 1 t N , M N → ∞ t_N,M_N\to\infty t N , M N → ∞ R N = t N + M N R_N=t_N+M_N R N = t N + M N
M N { e ( t N , M N ) + t N − 1 } → 0 , τ N , R N → 0. (104) \sqrt{M_N}\left\{e(t_N,M_N)+t_N^{-1}\right\}\to0,\qquad\tau_{N,R_N}\to0. \tag*{(104)} M N { e ( t N , M N ) + t N − 1 } → 0 , τ N , R N → 0. (104) For either trace centring, Z N , t N , M N ⋆ ⟹ 2 c d W Z^\star_{N,t_N,M_N}\Longrightarrow2\sqrt{c_d}W Z N , t N , M N ⋆ ⟹ 2 c d W D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) J 1 J_1 J 1 C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ])
M N ( F ^ N , t N , M N ⋆ − c d ) ⟹ N ( 0 , 4 c d ) . \sqrt{M_N}\left(\widehat{F}^\star_{N,t_N,M_N}-c_d\right)\Longrightarrow\mathcal{N}(0,4c_d). M N ( F N , t N , M N ⋆ − c d ) ⟹ N ( 0 , 4 c d ) . If also δ N , R N / M N → 0 \delta_{N,R_N}/\sqrt{M_N}\to0 δ N , R N / M N → 0 A N ⋆ ( t N + h ) 2 / t N A^\star_N(t_N+h)^2/t_N A N ⋆ ( t N + h ) 2 / t N E [ F ^ N , t N , M N ⋆ ] \mathbb{E}[\widehat{F}^\star_{N,t_N,M_N}] E [ F N , t N , M N ⋆ ]
Proof. Since e ( t , M ) = M / t e(t,M)=M/t e ( t , M ) = M / t M / t → 0 M/t\to0 M / t → 0 M 3 / 2 / t → 0 M^{3/2}/t\to0 M 3/2 / t → 0 Theorem 1.1 and Corollary 1.4 with κ = 0 \kappa=0 κ = 0 u = 1 u=1 u = 1 Corollary 10.6 below applies with ρ = 0 \rho=0 ρ = 0 δ N , R N / M → 0 \delta_{N,R_N}/\sqrt{M}\to0 δ N , R N / M → 0
For a non-bipartite base, the ideal common-time process S t , M P ( u ) = M − 1 / 2 ∑ h < ⌊ M u ⌋ ( P t + h 2 / t − c d ) S^P_{t,M}(u)=M^{-1/2}\sum_{h<\lfloor Mu\rfloor}(P^2_{t+h}/t-c_d) S t , M P ( u ) = M − 1/2 ∑ h < ⌊ M u ⌋ ( P t + h 2 / t − c d ) t , M → ∞ t,M\to\infty t , M → ∞ N N N Proposition 10.3 and (103) to the ideal squared traces to obtain
∥ S t , M P − S t , M P , ♯ ∥ ∞ ≤ c d M 3 / 2 2 t + 2 M t ∥ S t , M P , ♯ ∥ ∞ → 0 \left\|S^P_{t,M}-S^{P,\sharp}_{t,M}\right\|_\infty\le\frac{c_dM^{3/2}}{2t}+\frac{2M}{t}\left\|S^{P,\sharp}_{t,M}\right\|_\infty\to0 S t , M P − S t , M P , ♯ ∞ ≤ 2 t c d M 3/2 + t 2 M S t , M P , ♯ ∞ → 0 in probability. Interpolation preserves this bound. The finite-lift cycle comparison, also jointly with a time average and fixed low modes, and its fluctuation-scale remainder follow from § 9.2 with w h = 1 w_h=1 w h = 1
Expectation-centred squared traces
The two trace centrings need not centre their squares. To subtract the expectation of every squared observation, define
Z N , t , M ♯ , ⋆ , E ( u ) = Z N , t , M ⋆ ( u ) − E [ Z N , t , M ⋆ ( u ) ] , (105) Z^{\sharp,\star,\mathrm{E}}_{N,t,M}(u)=Z^\star_{N,t,M}(u)-\mathbb{E}[Z^\star_{N,t,M}(u)], \tag*{(105)} Z N , t , M ♯ , ⋆ , E ( u ) = Z N , t , M ⋆ ( u ) − E [ Z N , t , M ⋆ ( u )] , (105) Z N , t , M ⋆ , E ( u ) = Z N , t , M ⋆ ( u ) − E [ Z N , t , M ⋆ ( u ) ] . (106) Z^{\star,\mathrm{E}}_{N,t,M}(u)=Z^\star_{N,t,M}(u)-\mathbb{E}[Z^\star_{N,t,M}(u)]. \tag*{(106)} Z N , t , M ⋆ , E ( u ) = Z N , t , M ⋆ ( u ) − E [ Z N , t , M ⋆ ( u )] . (106) Thus the summands in (105) are { A N ⋆ ( t + h ) 2 − E [ A N ⋆ ( t + h ) 2 ] } / ( t + h ) \{A^\star_N(t+h)^2-\mathbb{E}[A^\star_N(t+h)^2]\}/(t+h) { A N ⋆ ( t + h ) 2 − E [ A N ⋆ ( t + h ) 2 ]} / ( t + h ) M − 1 / 2 M^{-1/2} M − 1/2 (106) the denominator is t t t (92) separately at every observation, we sum the local moment errors first.
Corollary 10.6 (Expectation-centred functional limits). Assume (2) and, in addition,
δ N , R N M N ⟶ 0. (107) \frac{\delta_{N,R_N}}{\sqrt{M_N}}\longrightarrow0. \tag*{(107)} M N δ N , R N ⟶ 0. (107) Set σ B = 2 β B c d \sigma_B = 2\sqrt{\beta_B c_d} σ B = 2 β B c d
Z N , t N , M N ♮ , ⋆ , E ⟹ σ B W in D ( [ 0 , 1 ] ) with J 1 , Z_{N,t_N,M_N}^{\natural,\star,\mathrm{E}} \Longrightarrow\sigma_B W \quad\text{in } D([0,1]) \text{ with } J_1, Z N , t N , M N ♮ , ⋆ , E ⟹ σ B W in D ([ 0 , 1 ]) with J 1 , and the polygonal interpolations converge in C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) Theorem 1.1 , for both choices of trace centring, are uniformly close in probability and converge jointly to the same Brownian motion. This convergence holds jointly with every fixed vector ( Y N , k ) k = 3 L (Y_{N,k})_{k=3}^{L} ( Y N , k ) k = 3 L W W W
If also M N / t N → ρ ∈ [ 0 , ∞ ) M_N/t_N \to\rho\in[0,\infty) M N / t N → ρ ∈ [ 0 , ∞ )
Z N , t N , M N ⋆ , E ( u ) ⟹ σ B ∫ 0 u ( 1 + ρ v ) d W ( v ) Z_{N,t_N,M_N}^{\star,\mathrm{E}}(u) \Longrightarrow\sigma_B \int_0^u (1+\rho v)\,\mathrm{d}W(v) Z N , t N , M N ⋆ , E ( u ) ⟹ σ B ∫ 0 u ( 1 + ρ v ) d W ( v ) in D ( [ 0 , 1 ] ) D([0,1]) D ([ 0 , 1 ]) J 1 J_1 J 1 C ( [ 0 , 1 ] ) C([0,1]) C ([ 0 , 1 ]) Corollary 1.3 , with the same Brownian motion, for both trace centrings and with the same independent limiting low-mode vector. In particular,
M N ( F ^ N , t N , M N ♮ , ⋆ − E [ F ^ N , t N , M N ♮ , ⋆ ] ) ⟹ N ( 0 , 4 β B c d ) , \sqrt{M_N}\left(\widehat{F}_{N,t_N,M_N}^{\natural,\star}-\mathbb{E}\left[\widehat{F}_{N,t_N,M_N}^{\natural,\star}\right]\right)\Longrightarrow\mathcal{N}(0,4\beta_B c_d), M N ( F N , t N , M N ♮ , ⋆ − E [ F N , t N , M N ♮ , ⋆ ] ) ⟹ N ( 0 , 4 β B c d ) , M N ( F ^ N , t N , M N ⋆ − E [ F ^ N , t N , M N ⋆ ] ) ⟹ N ( 0 , 4 β B c d ( 1 + ρ + ρ 2 / 3 ) ) , \sqrt{M_N}\left(\widehat{F}_{N,t_N,M_N}^{\star}-\mathbb{E}\left[\widehat{F}_{N,t_N,M_N}^{\star}\right]\right)\Longrightarrow\mathcal{N}\left(0,4\beta_B c_d(1+\rho+\rho^2/3)\right), M N ( F N , t N , M N ⋆ − E [ F N , t N , M N ⋆ ] ) ⟹ N ( 0 , 4 β B c d ( 1 + ρ + ρ 2 /3 ) ) , where the second conclusion uses M N / t N → ρ M_N/t_N \to\rho M N / t N → ρ
Proof. Write t = t N t=t_N t = t N M = M N M=M_N M = M N R = t + M R=t+M R = t + M H = H t , M H=H_{t,M} H = H t , M N ≥ 8 R 2 N\geq8R^2 N ≥ 8 R 2 3 ≤ s < R 3\leq s<R 3 ≤ s < R (72) with cutoff s s s (79) , gives
∣ E [ ( A N ⋆ ( s ) ) 2 ] s − E [ P s 2 ] s ∣ ≤ C B { s 5 q s N + 1 { ⋆ = ∗ } s 7 q s N 2 } . (108) \left|\frac{\mathbb{E}[(A_N^\star(s))^2]}{s}-\frac{\mathbb{E}[P_s^2]}{s}\right|
\leq C_B\left\{\frac{s^5q^s}{N}+\mathbf{1}_{\{\star=\ast\}}\frac{s^7q^s}{N^2}\right\}. \tag*{(108)} s E [( A N ⋆ ( s ) ) 2 ] − s E [ P s 2 ] ≤ C B { N s 5 q s + 1 { ⋆ = ∗ } N 2 s 7 q s } . (108) Indeed, exact trace centring leaves only the variance difference. For explicit trace centring, the extra term is { E [ Tr U G N S ] − m N ( s ) } 2 / s \{\mathbb{E}[\operatorname{Tr}U_{G_N}^{S}]-m_N(s)\}^2/s { E [ Tr U G N S ] − m N ( s ) } 2 / s s = 1 , 2 s=1,2 s = 1 , 2 P s P_s P s (68) implies
∣ E [ P s 2 ] s − c d η B ( s ) ∣ ≤ C B s , s ≥ 1. \left|\frac{\mathbb{E}[P_s^2]}{s}-c_d\eta_B(s)\right|\leq\frac{C_B}{s},\qquad s\geq1. s E [ P s 2 ] − c d η B ( s ) ≤ s C B , s ≥ 1. Here η B \eta_B η B (63) . For either period and every integer 0 ≤ n ≤ M 0\leq n\leq M 0 ≤ n ≤ M
∣ ∑ h = 0 n − 1 { η B ( t + h ) − 1 } ∣ ≤ 1. \left|\sum_{h=0}^{n-1}\{\eta_B(t+h)-1\}\right|\leq1. h = 0 ∑ n − 1 { η B ( t + h ) − 1 } ≤ 1. In the bipartite case this is a partial alternating sum; its cancellation is retained before taking absolute values.
Put b N , t , M ⋆ ( u ) = E [ Z N , t , M ♮ , ⋆ ( u ) ] b^\star_{N,t,M}(u)=\mathbb{E}[Z^{\natural,\star}_{N,t,M}(u)] b N , t , M ⋆ ( u ) = E [ Z N , t , M ♮ , ⋆ ( u )] (108) , bounding every partial sum by the full sum of absolute values gives
∥ b N , t , M ⋆ ∥ ∞ ≤ C B ( 1 + H ) M + C B N M ∑ s = t t + M − 1 s 5 q s + C B 1 { ⋆ = ∗ } N 2 M ∑ s = t t + M − 1 s 7 q s \left\|b^\star_{N,t,M}\right\|_\infty
\leq\frac{C_B(1+H)}{\sqrt{M}}+\frac{C_B}{N\sqrt{M}}\sum_{s=t}^{t+M-1}s^5q^s+\frac{C_B\mathbf{1}_{\{\star=\ast\}}}{N^2\sqrt{M}}\sum_{s=t}^{t+M-1}s^7q^s b N , t , M ⋆ ∞ ≤ M C B ( 1 + H ) + N M C B s = t ∑ t + M − 1 s 5 q s + N 2 M C B 1 { ⋆ = ∗ } s = t ∑ t + M − 1 s 7 q s ≤ C B { 1 + H M + δ N , R M ( 1 + 1 { ⋆ = ∗ } R 2 N ) } . (109) \leq C_B\left\{\frac{1+H}{\sqrt{M}}+\frac{\delta_{N,R}}{\sqrt{M}}\left(1+\mathbf{1}_{\{\star=\ast\}}\frac{R^2}{N}\right)\right\}. \tag*{(109)} ≤ C B { M 1 + H + M δ N , R ( 1 + 1 { ⋆ = ∗ } N R 2 ) } . (109) The geometric sums are taken before their terminal-cutoff bounds are used. Since H ≤ 1 + log M H \le1+\log M H ≤ 1 + log M τ N , R → 0 \tau_{N,R} \to0 τ N , R → 0 R 2 / N → 0 R^2/N \to0 R 2 / N → 0 Theorem 1.1 and Slutsky’s theorem. Interpolation is linear and preserves the uniform error bound.
For the common-time process suppose M / t → ρ < ∞ M/t \to\rho< \infty M / t → ρ < ∞ B j = b N , t , M ⋆ ( j / M ) B_j=b^\star_{N,t,M}(j/M) B j = b N , t , M ⋆ ( j / M )
∑ h = 0 n − 1 ( 1 + h / t ) ( B h + 1 − B h ) = ( 1 + ( n − 1 ) / t ) B n − 1 t ∑ j = 1 n − 1 B j , n = ⌊ M u ⌋ , \sum_{h=0}^{n-1}(1+h/t)(B_{h+1}-B_h)=\left(1+(n-1)/t\right)B_n-\frac{1}{t}\sum_{j=1}^{n-1}B_j,\qquad n=\lfloor Mu\rfloor, h = 0 ∑ n − 1 ( 1 + h / t ) ( B h + 1 − B h ) = ( 1 + ( n − 1 ) / t ) B n − t 1 j = 1 ∑ n − 1 B j , n = ⌊ M u ⌋ , with both sides zero for n = 0 n=0 n = 0 ( 1 + 2 M / t ) ∥ b N , t , M ⋆ ∥ ∞ (1+2M/t)\lVert b^\star_{N,t,M}\rVert_\infty ( 1 + 2 M / t ) ∥ b N , t , M ⋆ ∥ ∞ M / t M/t M / t Corollary 1.3 now proves the common-time limit and all the joint conclusions. The interpolated centring error has the same bound. Evaluation at u = 1 u=1 u = 1
Condition (10.15) is sufficient and is not claimed necessary. For example, along the sheet numbers N n = ( n 2 + n ) q n 2 + n N_n=(n^2+n)q^{n^2+n} N n = ( n 2 + n ) q n 2 + n t n = n 2 t_n=n^2 t n = n 2 M n = n M_n=n M n = n R n = n 2 + n R_n=n^2+n R n = n 2 + n δ N n , R n = 1 \delta_{N_n,R_n}=1 δ N n , R n = 1 τ N n , R n = 1 / R n \tau_{N_n,R_n}=1/R_n τ N n , R n = 1/ R n δ N n , R n / M n = n − 1 / 2 → 0 \delta_{N_n,R_n}/\sqrt{M_n}=n^{-1/2}\to0 δ N n , R n / M n = n − 1/2 → 0 δ N , R \delta_{N,R} δ N , R M / t → ρ < ∞ M/t\to\rho<\infty M / t → ρ < ∞ M 3 / 2 / t → 0 M^{3/2}/t\to0 M 3/2 / t → 0
Conclusions and spectral interpretation
A single-time observation and a time average have different limits. For a non-bipartite base, Theorem 1.5 and the continuous mapping theorem give
∣ A N ( t N ) ∣ 2 t N → d c d χ 1 2 . \frac{\lvert A_N(t_N)\rvert^2}{t_N}\xrightarrow{\mathrm{d}}c_d\chi_1^2. t N ∣ A N ( t N ) ∣ 2 d c d χ 1 2 . Thus the normalised squared trace retains randomness at a single growing time. In contrast, Theorem 1.2 recovers c d c_d c d Theorem 1.1 describes the cumulative error by 2 β c d W 2\sqrt{\beta}c_dW 2 β c d W M / t M/t M / t m N m_N m N 8 c d 8c_d 8 c d 4 c d 4c_d 4 c d
The common-time normalisation separates a different deterministic profile from the fluctuations. For M / t → ρ < ∞ M/t\to\rho<\infty M / t → ρ < ∞ Corollary 1.3 gives the variance clock u + ρ u 2 + ρ 2 u 3 / 3 u+\rho u^2+\rho^2u^3/3 u + ρ u 2 + ρ 2 u 3 /3 M / t → 0 M/t\to0 M / t → 0 Corollary 1.4 . Table 1 summarises the corresponding centring and scaling choices.
Table 1. Time normalisation, centring, and the resulting process limit.
Covariance and the Grover band. The covariance is determined by the band [ α d , π − α d ] [\alpha_d,\pi-\alpha_d] [ α d , π − α d ] ∣ ϕ ′ ∣ \lvert\phi'\rvert ∣ ϕ ′ ∣ θ = ϕ ( x ) \theta=\phi(x) θ = ϕ ( x ) Γ d \Gamma_d Γ d π − 1 1 I d ( θ ) d θ \pi^{-1}\mathbf{1}_{\mathcal{I}_d}(\theta)\,\mathrm{d}\theta π − 1 1 I d ( θ ) d θ I d \mathcal{I}_d I d Lemma 7.4 . Reflection symmetry makes this transform real; invariance under a shift by π \pi π c 3 ≈ 1.567306208 c_3\approx1.567306208 c 3 ≈ 1.567306208 Γ 3 ( 2 ) ≈ − 0.400140585 \Gamma_3(2)\approx-0.400140585 Γ 3 ( 2 ) ≈ − 0.400140585 c 4 = 4 / 3 c_4=4/3 c 4 = 4/3 Γ 4 ( 2 ) = − 3 / π \Gamma_4(2)=-\sqrt{3}/\pi Γ 4 ( 2 ) = − 3 / π
Relation to the orthogonal covariance form. For real polynomials F , G F,G F , G F ( 2 cos x ) = c 0 ( F ) + ∑ k ≥ 1 c k ( F ) cos ( k x ) F(2\cos x)=c_0(F)+\sum_{k\geq1}c_k(F)\cos(kx) F ( 2 cos x ) = c 0 ( F ) + ∑ k ≥ 1 c k ( F ) cos ( k x ) [6 Section 1.3.1 and Theorem 2.5] , the leading orthogonal covariance form is
Q O ( F , G ) = 1 2 ∑ k ≥ 1 k c k ( F ) c k ( G ) . Q_{\mathrm{O}}(F,G)=\frac{1}{2}\sum_{k\geq1}kc_k(F)c_k(G). Q O ( F , G ) = 2 1 k ≥ 1 ∑ k c k ( F ) c k ( G ) . For F s ( y ) = 2 T s ( r d y / 2 ) F_s(y)=2T_s(r_dy/2) F s ( y ) = 2 T s ( r d y /2 ) c k ( F s ) = 2 a s , k c_k(F_s)=2a_{s,k} c k ( F s ) = 2 a s , k
Q O ( F s , F u ) = 2 ∑ k ≥ 1 k a s , k a u , k . Q_{\mathrm{O}}(F_s,F_u)=2\sum_{k\geq1}ka_{s,k}a_{u,k}. Q O ( F s , F u ) = 2 k ≥ 1 ∑ k a s , k a u , k . Theorem 6.4 evaluates this form for all positive observation times. The period correction and finite-lift estimates justify the use of this form for the present ensemble. Transferring a Wigner-matrix limit theorem would require more than equality of the covariance forms. The Grover determinant and old/new decomposition remain established algebraic inputs [1 , 21 , 36 ] .
Time scale and scope. The evolution operator acts on a space of dimension D N = 2 m 0 N D_N=2m_0N D N = 2 m 0 N t N / D N → 0 t_N/D_N\to0 t N / D N → 0
F N c ( t N ) / D N = ( c d + o ( 1 ) ) t N / D N → 0. \mathcal{F}^{c}_{N}(t_N)/D_N=(c_d+o(1))t_N/D_N\to0. F N c ( t N ) / D N = ( c d + o ( 1 )) t N / D N → 0. The linear law concerns original eigenphases without unfolding, at times small compared with the matrix dimension. All finite-lift limits are taken under the unconditioned covering-permutation law. They neither assert almost-sure convergence under a coupling across sheet numbers nor use the sampling conditional on connectedness in [1 ] .
The squared trace is also distinct from an average return probability. With δ e \delta_e δ e e e e ℓ 2 ( D ( G ) ) \ell^2(D(G)) ℓ 2 ( D ( G ))
∣ ∑ e ⟨ δ e , U G s δ e ⟩ ∣ 2 ≠ ∑ e ∣ ⟨ δ e , U G s δ e ⟩ ∣ 2 . \left|\sum_e\langle\delta_e,U_G^s\delta_e\rangle\right|^2\ne\sum_e\left|\langle\delta_e,U_G^s\delta_e\rangle\right|^2. e ∑ ⟨ δ e , U G s δ e ⟩ 2 = e ∑ ∣ ⟨ δ e , U G s δ e ⟩ ∣ 2 . These conclusions concern spectral statistics rather than position-space transport, localisation, mixing, or local eigenvalue universality. The cutoff conditions are sufficient, not asserted optimal. Proposition 9.4 gives an ideal-model variance limit; the finite-lift weak limits do not by themselves yield convergence of the scaled variance or fourth moments. Expectation-centred versions of both squared-trace processes are established in Corollary 10.6 under the additional sufficient condition δ N , R / M → 0 \delta_{N,R}/\sqrt{M}\to0 δ N , R / M → 0 M / t → ρ < ∞ M/t\to\rho<\infty M / t → ρ < ∞
Statements and Declarations
Funding. This work was supported by JSPS KAKENHI Grant Number 23K12507.
Competing interests. The authors declare no conflicts of interest.
Data availability. The results are theoretical and are established by the proofs in this article. No external datasets are required for these results.
